Past Projects

Project Description:

Attributed graphs, having side information from nodes and edges, are widely used in many fields within the DOE applications, such as neuroscience, biological discovery, power grid, and distributed computing facilities. Learning the similarities of graphs, also known as graph matching, is one of the fundamental problems in machine learning tasks with structured data. Even though the graph-matching problem has been studied in the last decades, the research on learning the similarities between attributed graphs still remains open. Moreover, methods inspired by optimal transport, such as Gromov-Hausdorff and (Fused) Gromov-Wasserstein distances, have shown promising results to compare not only on structures but also in attributed graphs.

To this end, in our project, we will focus on designing new algorithms for comparing attributed graphs and developing of optimization algorithm to provide reliable metrics for attributed graph comparison. We will leverage recent updates in optimal transport in structured data, such as POT (https://pythonot.github.io/), to develop new algorithm handling with graph data, and apply them to a wide range of machine learning models, including the classical machine learning and deep learning approaches.

Disciplines: Applied Mathematics, Combinatorics, Geometry, and Topology

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL or virtual

Project Description:

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA

Project Description:

Disciplines: Analysis, and Applied Mathematics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL

Project Description:

Disciplines: Analysis, and Applied Mathematics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL

U.S. Citizenship is a requirement for this internship

Project Description:

Student will utilize spectrogram images/digital data to identify patterns that indicate the passage of watercraft.  An extensive suite of vessel wake data is available to develop robust training algorithms as well as sample data to verify and develop a vessel wake detection algorithm.  Student should possess working knowledge of ML concepts and be able to work independently in MATLAB and/or Python environment.  Experience with data analysis, including digital filtering, wavelet analysis and higher level ML tools/applications is highly desirable. Work will mostly be in an office setting but some possibility for field work during vessel wake collections for interested students.

Disciplines: Analysis, Applied Mathematics, Mathematics (General), Operations Research, Probability and Statistics, and Topology

Hosting Site:

U.S. Army Corps of Engineers, Engineer Research and Development Center (ERDC)

Internship location: Vicksburg, MS

Project Description:

In this project, novel deep learning algorithms will be constructed to learn solutions to the Fokker-Planck equations for stochastic dynamical systems. The key challenges to overcome include the possibility of non-locality, i.e., when such systems are driven by Levy noise; high-dimensionality, and non-Markovian characteristics. Potential data-driven solutions to such systems include the use of normalizing flows, generative adversarial networks, and neural stochastic differential equations. Some preliminary work in this area has been done by our team (across ANL, IIT-Chicago, Johns Hopkins University) here: https://arxiv.org/pdf/2107.13735.pdf

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL

Project Description:

Deep learning models are gaining wide interest within the scientific computing community due to their role in detecting and explaining underlying correlations between the different physical quantities that characterize a complex system. In particular, graph convolutional neural networks (GCNNs) have been showing great potential to describe the behavior of materials at microscopic scale by accurately capturing and describing interatomic interactions.

The predictive performance of GCNNs is very sensitive to the choice of the architecture for multiple hyperparameters such as the number of neurons per layers, the number of convolutional layers, the number of fully connected layers, the radius cutoff, the activation functions at each hidden layer, the learning rate and the batch size to iteratively train the model. All these hyperparameters strongly impact the predictions made by a GCNN model and GCNNs with different hyperparameter setups may produce vastly different predictions for the same input data. In particular, some choice of hyperparameters may lead to a poor predictive performance due to numerical artifacts such as overfitting or underfitting. Therefore, identifying an appropriate setting of hyperparameters is essential to ensure the model’s accuracy and generalizability.

Identifying a hyperparameter configuration that would make GCNN both accurate and robust requires performing an exhaustive search over a high dimensional space, which, in general, is computationally expensive. High performance computing can be leveraged to alleviate the computational burden of hyperparameter optimization (HPO) by concurrently exploring several hyperparameter configurations with distributed computing resources.

In this work, we will develop and implement scalable HPO algorithms for GCNNs. We will use the RayTune library for hyperparameter tuning and we will integrate the RayTune functionalities into an existing implementation of GCNNs. The performance of the HPO procedure will be assessed in terms of: (1) scalability attained by distributing the hyperparameter search over hundreds of compute nodes on supercomputers at the Oak Ridge Leadership Computing Facility (OLCF) and (2) validation accuracy of the trained GCNN model on ab-initio density functional theory (DFT) data generated by material scientists at Oak Ridge National Laboratory (ORNL) that describe the functional behavior of solid solution alloys at atomic scale.

The expected outcome is a scalable HPO framework integrated with the existing implementation of the GCNN model that attains linear scaling up to 100 compute nodes on the OLCF supercomputer Summit, with an improved accuracy by a factor of 10x with respect to existing GCNN models trained on the DFT data.

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

• The student is required to have preliminary fundamental knowledge about deep learning concepts and PyTorch.
• During the internship, the student is expected to regularly interact with strongly motivated research staff members at ORNL who develop ANN models for scientific applications.
• The student is expected to familiarize with the fundamental concepts of ANN models.
• The student is expected to familiarize and assimilate basic soft skills and apply them in a dynamic environment within a team of highly motivated scientists.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

U.S. Citizenship is a requirement for this internship

Project Description:

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

Project Description:

This summer experience revolves around algorithm creation and analysis for design problems for which goals are only available through observations of complex systems. Such settings commonly arise in the fields of derivative-free (zeroth-order) optimization. Our objectives include some combination of modeling specific problems, implementing novel algorithms, and analyzing (non)asymptotic performance.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA or virtual

Project Description:

• The student will learn basic concepts of graph theory, along with software capabilities provided in PyTorch and PyTorch-Geometric.
• The student will be exposed to high-performance computing (HPC) challenges that involve deploying GNN models on large scale platforms for incorporation in production codes.
• The student will familiarize with state-of-the-art challenges in atomic modeling for material science applications.
• The student will also develop presentation skills by describing the technical outcomes of his work to the ORNL community.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

Our group has decades of experience on developing and using automatic differentiation, which is known as back-propagation or autodiff in the Machine Learning frameworks. We are developing alternatives to back-propagation that take a more flexible approach on how to compute gradients, inspired by techniques developed in the context of differential equations and related problems.
In this project, you would help us develop new methods for gradient computations, and apply them to a variety of neural networks. If successful, this research could enable faster and more energy-efficient ways of training neural networks on GPUs, AI accelerator hardware, and other devices.

Disciplines: Applied Mathematics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL

Project Description:

Our group has developed methods for mapping the evaluation of certain mathematical functions to modern processors, for example by exploiting the associativity of operators to allow dynamic scheduling and accumulation of results. This allows us to compute these functions faster and using less energy.
In this project, you would help us develop the theory and software to perform these computations even faster, more reliably, or on different hardware platforms including GPUs or other accelerators. If successful, this work can improve the building blocks that are used by developers of scientific computing, engineering, and machine learning applications.

Disciplines: Applied Mathematics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL

U.S. Citizenship is a requirement for this internship

Project Description:

We'll be assisting with perspectives and interpretations applying differential geometry and competitive linear and non-linear model-based dimension reductions to explore applications ranging from computer vision to materials science and next generation communications technology. Example applications include: statistics of material microstructures, general image classification, and high-dimensional bi-criteria optimization for next generation wireless spectrum sharing. The position requires a candidate curious to explore some or all of the following topics in applied mathematics: (i) novel low-dimensional visualization and approximation methods, (ii) abstractions of computational differential geometry over matrix manifolds, (iii) linear and non-linear model-parameter dimension reduction, and (iv) infinite dimensional extensions of spaces of discrete shapes.

Disciplines: Applied Mathematics, Geometry, Probability and Statistics, and Topology

Hosting Site:

National Institute of Standards and Technology (NIST)

Internship location: Boulder, CO

Project Description:

Wildland fire is a natural process that has become problematic in society because of the expansion of human developments, increased fuel loads due to past fire suppression activities, climate change, and a myriad of other factors. Solutions for this problem require a more advanced understanding of the fundamental physical processes of these fires and how they propagate from the very small scale (fuel particles) to landscapes. Large efforts are currently underway to integrate highly instrumented field experiments, machine learning, artificial intelligence, and computational fluid dynamics models to advance our decision making in the future.

The applicant, with the guidance of several mentors, will have the opportunity to design an experience that focuses on their analytical strengths to help us to disentangle and understand complex relationships of fire spread and behavior. The applicant will examine a set (n=30) of recent fire field experiments with data including multi-temporal 3-D laser scanning (LiDAR), infrared and color video, 3-D wind fields, temperature profiles, and radiative fluxes. The primary objectives of the experience are: 1) Expand the applicant’s understanding of datasets of different spatial and temporal resolutions, 2) develop an approach to decompose and relate these data streams, and 3) to present the techniques and results in a way that is understandable to scientists from other disciplines and land managers.

This internship will be based at the Forestry Sciences Laboratory in Morgantown, WV in collaboration with Scientists from the USDA Forest Service, Rochester Institute of Technology, West Virginia University, and other institutions. The applicant will have the opportunity to collect data (in a learning setting) with the same instruments used in the fire experiments to understand their intricacies and limitations. Depending on Covid restrictions, the applicant may have the opportunity to visit several field sites, interact with other scientists and fire managers, and observe a prescribed burn.

Disciplines: Applied Mathematics, Probability and Statistics, and Topology

Hosting Site:

USDA Forest Service, Northern Research Station

Internship location: Morgantown, WV

U.S. Citizenship is a requirement for this internship

Project Description:

Disciplines: Analysis, and Probability and Statistics

Hosting Site:

U.S. Army Corps of Engineers, Cold Regions Research and Engineering Laboratory (CRREL)

Internship location: Fairbanks, AK

Project Description:

To enable practically useful quantum computing, this project aims to improve the error rate of logical gates on superconducting quantum devices by augmenting the quantum dynamical model with a data-driven approach to identify and incorporate latent dynamics. Quantum dynamics are often modeled by Schrödinger’s and Liouville-von Neumann’s equations, that evolve the quantum state and density matrix according to a Hamiltonian model. The Universal Differential Equations approach augments this model by a neural network that is trained from device data to account for unknown dynamics, such as drift in system parameters, control line losses, cross talk, and environmental interactions. The trained augmented model will provide a more accurate simulation of the quantum dynamics in superconducting quantum devices, that will then be used within our optimal control software stack to design control strategies that achieve higher fidelity of quantum gates on noisy quantum hardware. The trained neural network will further be analyzed using symbolic regression to provide a mathematical description and understanding of the latent quantum dynamics.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Livermore National Laboratory (LLNL)

Internship location: Livermore, CA

Project Description:

The goal of this project is to develop tools for image and data analysis, by leveraging scientific computing and machine learning algorithms. Various research opportunities exist in the following topics:
- Image segmentation, region and geometry detection in 2d/3d images,
- Analysis and statistics of geometries and shapes,
- Finite element modeling and meshing of evolving surfaces,
- Shape and topology optimization,
- Image-based meshing,
with applications in material science, biology and forensics. These projects involve knowledge of different mathematical areas, such as variational models, energy minimization, free boundary problems, meshing, mesh adaptivity and smoothing, continuous and discrete optimization, dynamic programming, and machine learning.

Disciplines: Applied Mathematics, Geometry, and Probability and Statistics

Hosting Site:

National Institute of Standards and Technology (NIST)

Internship location: Gaithersburg, MD

Project Description:

Large language models (LLMs) such as ChatGPT and LLaMA have had transformative impact on natural language processing, with significant scientific knowledge processing. Better understanding of LLMs may be necessary for further advances in this high interdisciplinary field. This project will involve learning about the LLMs and their applications, mathematical foundations, and explore further possibilities in applying LLMs for physics and other natural sciences, including processing the scientific knowledge information.

Disciplines: Applied Mathematics, Computational Mathematics, Geometry, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

Project Description:

Over the past three centuries, significant attention has been dedicated to the study of how mechanical instabilities such as wrinkling, buckling, and blistering cause patterns to emerge in inanimate materials. Can these principles be applied to the analysis of living systems? From the fractal-like edges of a kale leaf to the undulating crinkles of the human brain, many of the striking morphologies emblematic of living entities do not arise exclusively from gene expression programs but rather from the presence of mechanical forces. Elucidating this relationship between applied stresses and geometric forms will help guide scientific innovations ranging from the creation of bioinspired materials to the establishment of metrological standards. Students will gain experience with topics in applied and computational mathematics, particularly the calculus of variations, differential geometry, and nonlinear partial differential equations.

Disciplines: Applied Mathematics, and Computational Mathematics, Mathematical Biology Analysis

Hosting Site:

National Institute of Standards and Technology (NIST)

Internship location: Gaithersburg, MD or virtual

Project Description:

The purpose of this project is to study large language models (LLMs) that are founded on explicit probabilistic models, rather than on transformer-type architectures. The objective is to obtain predictive models for discrete sequence data such as natural language, DNA sequences, chemical formulae, etc. that are transparent in their operation and amenable to furnishing characterizations of predictive uncertainty.

The MSGI Intern will participate in theoretical development and exploration of models, and in exploration of corpuses of sequential data to determine model structures appropriate to each type of data. As this is a computational statistics project, the intern should expect to write code implementations of models, and also to code data exploration methods.  Equally important is the expectation that the intern will prepare a report that should function as an outline or first draft of a paper on the methods developed in the project.

The intern will learn techniques for probabilistic modeling of discrete sequences, and for characterizing the statistical behavior of this type of data. A model in current development uses Gaussian process (GP)-type methods, so the intern should expect to learn the theory and practice of GPs. Model training is likely to require use of Argonne's high-performance computing (HPC) facilities, and the intern should also expect to learn to utilize such facilities.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL

Project Description:

Expensive-to-evaluate stochastic oracles appear throughout the quantum information sciences research field. Unfortunately, optimizing such oracles is considerably more difficult than optimizing deterministic oracles. This is because of the varying output observed for repeated calls to the oracle, even with a fixed set of input variables. This project seeks to develop, analyze, and implement numerical methods for optimizing oracles that are stochastic in nature and that are relatively expensive to evaluate. Application problems include stochastic oracles that appear across the quantum information sciences. We are especially interested in using (polynomial and Bayesian) model-based methods to identify high-quality local optima for such problems.  

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL

Project Description:

The project aims to explore and implement mathematical computations for causality representing learning in scientific data leveraging deep-learning models like transformers, graph neural networks.

Causality in scientific data mostly denotes what process, event, or state of a variable may cause a change of that event or state of one/multiple variables in low/high fidelity scientific data. Causality Learning is a challenging task for many scientific domains due to high dimensional data, unobserved or partially observed physics knowledge, and the availability of limited ground truth. Learning causality or dependency structure among different features (e.g., particle-particle interactions in fluids, neuron connectivity in neuroscience, etc.) that can be directly obtained from low-fidelity scientific simulations can aid in modeling and understanding high-fidelity data.

 Student Responsibilities: The student is expected to have a foundational knowledge of training deep learning models with Pytorch, Python data analysis and visualization libraries pandas, numpy, scikit-learn, matplotlib. The student is expected to implement and test transformer or graph neural networks and test the performance of such models on open-source scientific datasets. The student will collaborate with ORNL staff to validate the performance of causality and address multiple scientific communities at ORNL.

Learning objectives:

1. The student will learn basic concepts about causality and state-of-the-art machine learning models to learn causality.
2. The student will familiarize with training deep learning architectures and the mathematical computations involved with the popular models, like transformers, graph neural networks.
3. The student will be exposed to the problems and challenges of learning causality for different scientific applications, e.g., neuroscience, climate, etc.
4. The student will develop presentation skills by describing the technical outcomes of his/her research to the ORNL community.

Disciplines: Analysis, and Applied Mathematics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

Dataset imbalance refers to the issue when certain classes are represented by significantly more number of data points relative to others. It is a prevalent issue in machine learning especially classification problems in many scientific applications. This issue materializes itself when the final performance of a model is biased towards the class with a larger number of sample points. One way to correct this bias is to equalize the imbalance and intelligent sampling strategies play a critical role in this procedure. However, due to a lack of efficient approaches, a common way to address the issue involves trial and error driven uniform oversampling of the underrepresented class or undersampling of the over-represented class.

In this project, we will formulate the problem of imbalance in a data batch as an optimization problem and derive conditions which must be satisfied for sampling a balanced data batch. We then integrate the condition into the neural network learning problem. We will develop a game theoretic approach to resolve the tradeoff between the performance of the neural network and the variance in the data.

Disciplines: Applied Mathematics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL

Project Description:

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

USDA Forest Service, Southern Research Station

Project Description:

Adversarial robustness is a pressing concern in multimodal frameworks, especially in large language models (LLMs) that solve a spectrum of problems like text-to-image, image-to-text, and text-to-text translations. Multimodel LLM such as ChatGPT, despite their powerful capabilities, are vulnerable to adversarial attacks. These attacks subtly manipulate input modalities, compromising model responses, and can even be insidiously embedded within training datasets, affecting model embeddings and overall performance. Building upon our previous work, where we utilized the AdversarialTensors framework to effectively mitigate adversarial noises in an unsupervised manner for standard CNNs, we aim to extend this ability to enhance the robustness of multimodal frameworks. Our goal is to purify input modalities from adversarial contaminations before they are processed by LLMs and to fortify model robustness by integrating tensorial defense strategies alongside LORA—a widely adopted, low-complexity fine-tuning model. Moreover, by integrating knowledge graphs, we can harness the structured and interlinked knowledge between input and corresponding embeddings to identify and counteract inconsistencies or adversarial perturbations, thereby enhancing the model's ability to reason and improve its resilience against malicious attacks.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM or virtual

Project Description:

This project aims to develop spatial mechanistic models of respiratory virus infection (such as SARS-CoV-2, influenza) and the immune responses to understand how stochasticity and immune regulation impact on viral infection and pathogenesis. Respiratory viruses (such as SARS-CoV-2 and influenza) cause a high mortality and morbidity every year. Their infections usually result in a wide range of clinical outcomes, from asymptomatic to lethal. There is an urgent need to quantitatively define the specific factors that govern infection outcomes to develop better therapeutic and intervention strategies. Therefore, the work will be critical for rational design of vaccines to prepare for the current or next pandemic caused by respiratory viruses.

Disciplines: Computational Mathematics, and Mathematical Biology

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

Project Description:

Matrix-free high-order finite element method (MF-FEM) is a cutting-edge method to perform fast, robust, and accurate simulations of advection-diffusion partial differential equations (PDEs) that are ubiquitous in computational physics. However, currently the application of MF-FEM is limited to algorithms that are explicit in time. The presence of diffusion or stiff source terms calls for implicit or implicit-explicit timestepping methods, which negatively impacts the efficiency of MF-FEM. To overcome this performance issue, we propose to couple fast multigrid (MG) methods for the discrete diffusion operators with MF-FEM for advection operators.

Geometric multigrid (GMG) methods are one of the fastest classes of linear and nonlinear solvers for implicit equations arising in numerical methods for PDEs. GMG methods typically rely on a hierarchy of nested structured meshes, and the physical equations are solved on ""coarse"" meshes to provide corrections to the solution on the original ""fine"" mesh. For unstructured meshes that are moving or adapting in time, it is not obvious how to construct a mesh hierarchy, and algebraic multigrid (AMG) methods provide a less intrusive multigrid option that are often used instead. However, AMG methods require assembling a full matrix for the discretization, and, although efficient, do not compete with GMG in terms of computational efficiency, particularly on GPUs. Some previous work has considered semi-geometric MG (sGMG) methods, where an initial simplicial (triangular) mesh is coarsened to construct a nested grid hierarchy on which to apply GMG. We are, however, interested in coupling implicit solvers to Arbitrary Lagrangian-Eulerian simulations of hydrodynamics, where quadrilateral and hexahedral (non-simplicial) meshes offer significantly higher physics fidelity.

The goals of the proposed project are: (1) the development of novel sGMG solvers for the advection-diffusion PDEs on quadrilateral and hexahedral meshes and (2) efficient implementation in LANL’s open source GPU-based code Fierro (https://lanl.github.io/Fierro).

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

Project Description:

Many complex systems are usually represented by networks (e.g., communication networks, power grids, social networks, etc.). Communities or clusters are key to understand the structure and function of these complex systems because communities represent important functional modules in networked systems. Therefore, there is an increasing interest in understanding the limits of the robustness of the community structure. This is because maintaining the functionality of networked systems is heavily dependent on preserving their community structure.

Given ORNL's expertise on modeling and simulation of complex systems using leadership computing facilities, this project will take advance of modern data science, machine learning, and network science techniques, or any technique of interest to the participant that could help on better understand the limits of the robustness of the community structure in interconnected systems. We are highly interested in exploring data clustering techniques that use graph embeddings for clustering both synthetic and empirical real-world networks. We will seek to understand the limitations of current graph embedding methods for clustering tasks and propose ways to alleviate these limitations.

This project will allow the participant to actively drive an exciting facet of an ongoing research project at ORNL, and have their contributions directly integrated into the Computer Science and Mathematics Division research priorities. A successful student has prior experience with data science techniques, machine learning, and network science, but is not expected to have deep experience with programming.

Notably, prior projects at ORNL by interns in this team have led to published papers at top interdisciplinary/applied math journals. Examples include:

• PLOS ONE: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0284077
• IEEE IoTJ: https://ieeexplore.ieee.org/abstract/document/10213280
• PRE: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.108.054302

Based on the findings here, we will also seek to publish a paper in a major data science/applied math venue with the participant as the lead author.

Feel free to reach out if questions.

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN or virtual

Project Description:

The promise of ubiquitous connectivity brought by emerging technologies (i.e., 5G/6G) will enable interconnected systems (e.g., Internet of things (IoT), smart cities, science facilities, among others) to work faster and more efficiently. However, ubiquitous connectivity comes at the expense of augmenting the attack surface and the risk of suffering from advanced cyber attacks. Detection and adaptation to such threats requires collecting, filtering, and analyzing event data of heterogeneous scale, speed, and modality. The design and deployment of trustworthy methods and algorithms that can accommodate volume, velocity, and variety is key for near real-time detection, continuous adaptation, and quick attack recovery is challenging and requires innovation. Thus, to address the emerging challenges of next generation of cyber threats, the goal of this project is to design and implement a suite of algorithms that allow near real-time detection of adaptive cyber attacks in interconnected systems.

Given ORNL's expertise in acquisition and processing of data for enterprise and cyber physical domains using leadership computing facilities along with leading researchers in applied mathematics, computer science, statistics, and cybersecurity, this project will take advance of modern data science, machine learning, or any technique of interest to the participant (including but not limited to multivariate time series analysis, clustering, graph mining, etc.) that could help to design and implement algorithms that will enable quick detection, adaptation, and recovery from sophisticated and evolving cyberattacks targeting modern interconnected systems. Learning objectives for the applicant include: (1) develop a basic understanding of novel cyberattacks affecting modern interconnected systems; (2) design and development of data-driven models for detecting advanced cyberattacks; (3) validate the efficacy of developed algorithms using empirical data and compare its performance against state-of-the-art classical approaches.

This project will allow the participant to actively drive an exciting facet of an ongoing research project at ORNL, and have their contributions directly integrated into the Computer Science and Mathematics Division research priorities. A successful participant has prior experience with data science techniques and machine learning but is not expected to have deep experience with programming.

Notably, prior projects at ORNL by interns in this team have led to papers published at top interdisciplinary/applied mathematics journals. Examples include:

• PLOS ONE: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0284077
• IEEE IoTJ: https://ieeexplore.ieee.org/abstract/document/10213280
• PRE: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.108.054302

Based on the findings here, we will also seek to publish a paper in a major data science/applied math venue with the participant as the lead author.

Feel free to reach out if questions.

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN or virtual

Project Description:

Kleene algebra with tests (KAT) is an algebraic technique for mathematical reasoning about the behavior of computer programs. In this project we will explore the semantic foundations of KAT as applied to the task of program verification.

Potential projects include:

• generalizing KAT's theory to discharge realistic program verification conditions;
• developing algorithms for converting between traditional verification input and a KAT representation;
• a prototype implementation of a KAT-based program verifier;
• a mechanization of KAT in a mathematical proof assistant such as Lean, Coq, Agda, etc;
• exploring the connections between first-order satisfiability modulo theories (SMT) logic and KAT logic;
• or others, depending on student interest.

Disciplines: Computational Mathematics, and Foundations

Hosting Site:

Lawrence Livermore National Laboratory (LLNL)

Internship location: Livermore, CA or virtual

Project Description:

The goal of this project is to investigate the use of machine learning tools for solving combinatorial optimization problems.  Such problems seek to optimize an objective function defined over a discrete set of possible choices, the size of which is typically too large to exhaustively search.  These problems are often approached using principled heuristics, techniques to reduce the search space, or approximate solution strategies.

In this project, the student will learn about optimization, machine learning, and the use of related software tools. The student will be exposed to applications in material science and enzyme engineering.  They will collaborate with research staff in mathematics and scientific computing and have the opportunity to present their research at the end of the summer.

Disciplines: Applied Mathematics, Combinatorics, and Computational Mathematics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

U.S. Citizenship is a requirement for this internship

Project Description:

The Army maintains stringent requirements for equipment in the Arctic operational environment and expresses concerns of risk to mission failure for sustaining the forces at temperatures down to -65^oF, wind speeds greater than 100 mph, and 25 lb/ft² snow load. However, these conditions may only be encountered on rare occasions, and in certain subregions of the Arctic.

In this new start ESTCP project, we explore refining the Arctic climatic zones characterization relative to these design thresholds. This includes identifying the circumpolar north climate breaks for environmental parameters to narrow the contingency, logistics and resources needed, and develop site-focused requirements for the Services to operate across the region. To accomplish this, we intend to include three climate characteristics (air temperature extremes, snow depth and wind speed limits) to the currently defined climate zones and conduct analysis against materiel and equipment exposure thresholds. This will provide mission critical data and information for operations managers to make better risk-informed decisions and to operate equipment, materiel, and force projection in the circumpolar region.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

U.S. Army Corps of Engineers, Cold Regions Research and Engineering Laboratory (CRREL)

Internship location: Hanover, NH or virtual

Project Description:

We are exploring new optimization methodologies to optimize with digital twins. Optimization methodologies are a critical component to make digital twins a reality. They are needed in the optimal design of experiments that integrate data acquisition with simulation, the optimal steering of processes or devices modeled as digital twins, and the connection to neighboring digital twins. In this project, we will explore the use of distributed optimization techniques to control a swarm of vehicles used to optimally acquire environmental data along different paths. The project combines the solution of PDEs with the design of experiments and optimal control. Our goal is to build a prototype model and computational simulation using python and Fenics.

Disciplines: Computational Mathematics, and Probability and Statistics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL or virtual

Project Description:

Viruses with segmented genomes such as Crimean Congo hemorrhagic fever virus (CCHFV), hantaviruses, and influenza can form chimeric viruses when an individual host becomes co-infected by distinct variants of a given virus. This reassortment of genomic segments is postulated to be an important aspect in the emergence and re-emergences of both old and new infectious diseases. The ultimate goal of this project is to answer a scientific question: can we determine if a virus experienced a genomic reassortment event in the past that increased its fitness by looking at contemporary genomic samples of that virus? The challenge is that there are no robust off-the-shelf methods that can infer the presence of genomic reassortment (see https://doi.org/10.1101/2023.09.20.558687 for our preprint on this issue). Further, while there are interesting candidate methods that might allow for robust maximum likelihood inference of genomic reassortment from phylogenetic trees, it is unlikely that those methods will have reasonable computational efficiency. This project seeks to address that issue by 1) building a mechanistic model to simulate both the transmission dynamics and the viral evolution of a segmented virus in an idealized population, 2) build a Deep Neural Network mapping the simulated phylogenetic trees to the model parameters such as increased contagiousness of the reassorted virus, and 3) infer the presence of relevant genomic reassortment using real genetic sequence data from CCHFV. Participants will gain hands-on experience in epidemiological research, enhance their mathematical modeling, statistical analysis, and machine learning skills, and prepare for future careers in academia, research, or related industries. This opportunity contributes to vital scientific discoveries and equips students with highly sought-after skills in computational biology and public health research.

Disciplines: Applied Mathematics, Computational Mathematics, and Mathematical Biology

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

Project Description:

This project will build models based on observed discrepancies between theory computer simulations and data at the Large Hadron Collider that fill in the physics behind these gaps.

The models will be developed using artificial intelligence (AI) with two distinguishing and novel features:  we will learn (1) corrections to an existing physics model to describe data using reweighting and (2) symbolic relationships between theory features to construct better physics models.  A mature version of the developed tools will, for example, significantly reduce the measurement uncertainties on the top quark mass and possibly aid in the discovery of a new, weakly-interacting particle.

We propose to complete the early stages of this project with following goals:

(1)        Using techniques such as weakly supervised machine learning to automatically identify collider data in the Rivet analysis library that is poorly described by theory predictions and has no obvious physics explanation

(2)        Developing an AI-based regression model to describe the gap between theory and data using the Pythia event generator

(3)        Constructing metamodels, using Simulation-based inference (SBI) for example, to determine correlations between the gap and the latent variables of the generator. These metamodels will be used to supplement the standard generator predictions and provide candidate physical explanations and descriptions of the data.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Fermi National Accelerator Laboratory (FNAL)

Internship location: Batavia, IL or virtual

Project Description:

Monte Carlo-based event generators are essential tools for analyzing and interpreting data from particle collision experiments. Currently, the physical evolution of a particle collision, and hence the computing algorithms used to make predictions, are serial. On the other hand, many of our computations in the future will be done on chips with GPUs. This project aims to explore how the physics algorithms could be restructured to exploit acceleration. As a first case, we would explore the parton shower algorithm.

Disciplines: Computational Mathematics

Hosting Site:

Fermi National Accelerator Laboratory (FNAL)

Internship location: Batavia, IL or virtual

Project Description:

Randomized algorithms, such as sketching, sparsification, and streaming, are powerful dimensionality reduction tools for analyzing large. The applications for DOE include computational linear algebra (such as solvers, preconditioners, and low-rank approximation) and graph analysis (e.g., graph partitioning, clustering, and graph learning). Many impressive advances in the theory of randomized algorithms have not been translated into practical demonstrations. This project aims to bridge this gap between theory and practice, and to produce high-quality software running on modern HPC hardware with demonstrations in application codes.

Disciplines: Computational Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA or virtual

Project Description:

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

We are developing efficient physics-informed neural network reduced order models (NNROMs) to accelerate complicated, large-scale physical simulations. Currently, our physics-informed NNROM can reduce the dimensionality of an advection-dominated 2D Burgers simulation to a latent space of 5 with a relative error with respect to the corresponding full order model of less than 1% and accelerate the full order model simulation by a factor of 10, which cannot be achieved by any machine-learning black box approach. We plan to extend the ROM to large-scale problems, such as advection-dominated hydrodynamics, transport problems, turbulence, and Rayleigh–Taylor instability simulations. We expect our NNROM will achieve a higher speed-up when it is applied to larger-scale problems.

A student participating in our research project will first learn what our NNROM can do for the 2D Burgers simulation and then extend it to a new physics problem by training an autoencoder neural network and implementing NNROM on more complex problems, such as shock-moving hydrodynamics, pore-collapse dynamics, particle transport, plasma physics, and earthquake inverse problems. Depending on the results, we will write a journal paper together. The experience of implementing NNROM will let the student to apply it to other problems, including those that may be part of the student’s Masters or PhD thesis.

Disciplines: Analysis, Applied Mathematics, Mathematics (General), Operations Research, and Probability and Statistics

Hosting Site:

Lawrence Livermore National Laboratory (LLNL)

Internship location: Livermore, CA

Project Description:

We are developing efficient latent-space dynamics identification (LaSDI) algorithms to accurately accelerate parametric and complex physical systems. The reduced space dynamics after compression are often much simpler than the corresponding full space dynamics. Therefore, various models can be fit to identify the hidden dynamics in the reduced space, which in turn can be used to predict system response to new input parameter. We have successfully applied the latent-space learning algorithm, so called LaSDI, to accurately accelerate various benchmark problems, such as advection equation, Burgers’ equation, and heat conduction problems.

A student participating in our research project will first learn our existing tool box, LaSDI and gLaSDI. Then he or she will further improve LaSDI by exploiting other latent space model and extend it to more complex problems, such as shock-moving hydrodynamics, pore-collapse dynamics, particle transport, plasma physics, and earthquake inverse problems. Depending on the results, we will write a journal paper together. Our LaSDI is application-agnostic, so by the end of summer, the student will be able to apply the improved LaSDI method to a broad range of physical simulations, including those that may be part of the student’s Masters or PhD thesis.

Disciplines: Analysis, Applied Mathematics, Mathematics (General), Operations Research, and Probability and Statistics

Hosting Site:

Lawrence Livermore National Laboratory (LLNL)

Internship location: Livermore, CA

Project Description:

The Earth system represents a sophisticated, non-linear dynamic system. Traditionally, climate modeling is conducted through extensive Earth System Models, which are comprehensive computational structures designed to simulate interactions among Earth's various elements. Due to their enormity, these models encompass large datasets, posing challenges for direct analysis.

In recent years, Koopman operator theory has emerged as a mathematically rigorous method for analyzing and modeling dynamic systems. It is increasingly favored for its data-driven approach. The focus of this summer's research project is to enhance modern Koopman techniques for a more effective characterization of the non-equilibrium and chaotic dynamics present in the Earth system. Our goals include developing and implementing innovative Koopman machine learning models tailored for Earth system applications.

Disciplines: Analysis, Applied Mathematics, Probability and Statistics, and Topology

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM or virtual

Project Description:

Quantum computing is a promising venue to solve many problems which are classically complex. It’s been demonstrated that some of the quantum algorithms have clear advantage over the classical algorithms. However, quantum errors prevent the scalability of the quantum computers. We will theoretically explore the various generalized quantum errors in qubits and qudits, and classical tensor network algorithms, and find the interplay between these errors. Time permitting, actual implementation of the errors on real quantum hardware and a simple classical tensor network algorithm could be implemented. For this position, prior knowledge about quantum computation, tensor networks are not needed, however an exposure could be helpful. Strong background in linear algebra and programming is preferred, although the candidate is free to choose a programming language to use. This project will be conducted in a team setting under the primary direction of researchers at Fermilab. The entire project may be done remotely, with frequent video meetings and the use of other communication tools (e.g., Slack, email), or we may offer housing onsite at Fermilab, based on availability.

Disciplines: Algebra or Number Theory, and Applied Mathematics

Hosting Site:

Fermi National Accelerator Laboratory (FNAL)

Internship location: Batavia, IL or virtual

U.S. Citizenship is a requirement for this internship

Project Description:

Battle damage assessment involves assessing the physical damage to buildings and infrastructure from satellite images taken before and after an event. This project has focused on developing automated methods of doing so using neural networks. However, lack of generalization from one geographic region to another, limited labeled data, and the need for a confidence measure make this a complex problem. An intern on this project will have the opportunity to explore one of these factors (or another related aspect). We are seeking candidates with experience in programming (preferably python) and an interest in applying mathematical or statistical principles to computer vision and machine learning.

Disciplines: Applied Mathematics

Hosting Site:

U.S. Army Corps of Engineers, Geospatial Research Laboratory (GRL)

U.S. Citizenship is a requirement for this internship

Project Description:

This project will explore visual analytics aspects of perspectives and interpretations applying Riemannian geometry and competitive linear and non-linear model-based dimension reductions in applications ranging from wind energy infrastructure and computer vision to materials science and next generation communications technology. Example applications include: statistics of material microstructures, image and time series classification, and bi-criteria optimization for next generation wind turbine design/measurement as well as coexistence in wireless spectrum sharing. Using scientific visualization package like ParaView, the project will attempt to visualize and analyze applied mathematics. The position requires a candidate curious to explore some or all of the following topics in applied mathematics: (i) novel low-dimensional (< 4) visualization and approximation methods, (ii) computational differential geometry over matrix manifolds, (iii) linear and non-linear dimension reduction, (iv) sensitivity analysis over Riemannian manifolds, and (v) regularized representations of planar curves and embedded surfaces.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

National Institute of Standards and Technology (NIST)

U.S. Citizenship is a requirement for this internship

Project Description:

The identification of fallow fields from remote sensing images is challenging because of their similar spectral properties to re-cultivation or transitional classes. However, the temporal changes in spectral properties in each field helps to identify if a field is fallow or not. Fallow fields are possible indicators of several socio-economic-climatic changes including drought, changes in policies, and war. For these reasons, the ability to detect where and when fields are transitioning to or from a fallow state, is imperative to mitigate potential conflict and humanitarian disasters. This project will implement a temporal segmentation algorithm to detect fallow fields using multispectral time series. The time series will then be enumerated to spatial segments, using the Segment Anything Model (SAM) or other open-source methods. The last step is to apply a temporal segmentation algorithm to the spatially enumerated time series to detect changes in the spectral signature, defining a conversion from cultivated to fallow.

The student should have a working knowledge of ML and experience using Google Earth Engine or preferably Python using geospatial datasets. The student does not need to be an expert coder, one outcome of the internship is that the student will strengthen their coding skills and familiarity with open-source libraries and their application to solve geospatial problems. What is more important, is that the student can participate independently and is self-motivated.

*This topic is flexible. Replace ‘fallow fields’ with any possible feature (forest disturbance, wildfires, wetlands, etc.) – what is important is to develop the Earth Engine or STAC Framework to handle remotely sensed timeseries data and research a topic that is of interest to you.

The intern can be virtual or onsite in Alexandria, VA. The mentors are remote.

Disciplines: Applied Mathematics, and Biometrics and Biostatistics

Hosting Site:

U.S. Army Corps of Engineers, Geospatial Research Laboratory (GRL)

Internship location: Alexandria, VA or virtual

U.S. Citizenship is a requirement for this internship

Project Description:

The selected intern will be placed within a research team focused on geospatial analytics and prediction. The team is comprised of several PhD and MS level scientists with backgrounds in satellite remote sensing, GIS, or the physical sciences. The intern should expect to gain valuable experience in remote sensing methods, applied machine learning, and cloud computing during the course of this internship project. Deliverables are a key aspect of a professional portfolio and enable research to be more impactful within the organization. Accordingly, the intern will be supported by the mentor to publish the findings from this project in either a peer-reviewed journal article within the remote sensing discipline, or in a governmental technical report which is published online by the organization. The specific project topic is flexible based on the interests and skills of the intern as long as it aligns within the general topic of geospatial analytics and prediction.

One possible research project idea would leverage Synthetic Aperture Radar (SAR) with multispectral imagery observations in order to gap fill cloud cover from satellite imagery and generate more complete time series of land surface observations. While some research has been done in this domain, more advanced deep learning algorithms, such as long short term memory networks, are needed to increase the accuracy of this project.

Ultimately, the output from this project would be impactful and directly incorporated into decision support tools and/or transition of technology to our Army Enterprise customers.

It should be noted that the primary mentor is remote, however, he is available for virtual support and guidance. Other co-mentors and interns will be located on-site.

Disciplines: Analysis, and Applied Mathematics

Hosting Site:

U.S. Army Corps of Engineers, Geospatial Research Laboratory (GRL)

U.S. Citizenship is a requirement for this internship

Project Description:

The selected intern will be placed within a research team focused on geospatial analytics and prediction. The team is comprised of several PhD and MS level scientists with backgrounds in satellite remote sensing, GIS, or the physical sciences. The intern should expect to gain valuable experience in remote sensing methods, applied machine learning, and cloud computing during the course of this internship project. Deliverables are a key aspect of a professional portfolio and enable research to be more impactful within the organization. Accordingly, the intern will be supported by the mentor to publish the findings from this project in either a peer-reviewed journal article within the remote sensing discipline, or in a governmental technical report which is published online by the organization. The specific project topic is flexible based on the interests and skills of the intern as long as it aligns within the general topic of geospatial analytics and prediction.

One research idea would use readily-available cloud computing platforms with time series of satellite imagery observations to quantify trends in land cover changes and disturbance events across broad geographic areas. Some specific inquiries to address would include: 1) How have changes to earth's land cover varied over time within protected conservation areas versus non-protected areas? 2) How accurately can we model building heights from satellite remote sensing imagery using multiple satellite sensor images and ancillary geospatial data? For both of these example ideas, geospatial datasets, ground truth, and a computing platform are all available for analysis. The mentors are available to guide the intern and provide technical assistance for those who may be unfamiliar with GIS or remote sensing. It is not expected that the intern have GIS or remote sensing experience.

Ultimately, the output from this project would be impactful and directly incorporated into decision support tools and/or transition of technology to our Army Enterprise customers.

The primary mentor is remote, however, he is available for virtual support and guidance. Other co-mentors and interns will be located on-site and available as well.

Disciplines: Analysis, and Applied Mathematics

Hosting Site:

U.S. Army Corps of Engineers, Geospatial Research Laboratory (GRL)

Project Description:

Continual learning (CL) is a field concerned with learning a series of inter-related tasks that are typically defined in the sense of either regression or classification. In recent years, CL has been studied extensively when these tasks are defined using Euclidean data – data, such as images, that can be described by a set of vectors in an n-dimensional real space. However, when the data corresponding to a CL task is nonEuclidean– data, such as graphs, point clouds or manifold, the notion of similarity in the sense of Euclidean metric does not hold. For instance, a graph is described by a tuple of vertices and edges and similarities between two graphs is not well defined through a Euclidean metric. Due to this fundamental nature of graphs, developing and demonstrating efficacy with CL for nonEuclidean data presents several theoretical and methodological challenges. In particular, CL for graphs requires explicit modelling of nonstationary behavior of vertices and edges and their effects on the learning problem. To obviate this necessity, in our prior research we have developed an adaptive dynamic programming viewpoint for CL with graphs. Here, we formulate a two-player sequential game between the act of learning new tasks (generalization) and remembering previously learned tasks (forgetting).

The goal of this new project is to apply the theoretical principles of graph continual learning to scientific applications and demonstrate that, the two player principle formulated theoretically will improve the performance. In particular, we will start with a literature survey and identify the state of the art methods and datasets where the graph continual learning is relevant. Next, we will formulate a comparative study between the present method and the state of the art. Finally, we will consider the spatio-temporal traffic analysis problem and demonstrate the applicability of the graph continual learning approach on the spatio-temporal problem.

The applicant will get exposed to the world of lifelong learning, which is becoming very popular in recent times. Moreover, the applicant will be able to understand the impact of theoretical principles on real applications and explore the influence of parameter choices on such applications. The internship will culminate in a research paper that will be potentially submitted to AAAI-2025.

Disciplines: Analysis, Applied Mathematics, and Probability and Statistics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL

Project Description:

Diffusion probabilistic models (DPMs) offer state-of-the-art performance for various tasks including high-fidelity image generation, image editing, text-to-image generation, voice synthesis and data compression. The sample quality of DPMs depend critically on the guided sampling technique, which requires solving the diffusion ODEs or SDEs defined by the data prediction model. In this project, we aim to develop efficient ODE solvers to accelerate DPMs. We will exploit the semi-linear structure of the model, investigate new time integration methods and analyze their convergence and stability properties. The participant will obtain hands-on experience with diffusion models and high-performance ODE solvers with full GPU support.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL

Project Description:

The goal of this project is to design and analyze numerical methods for modeling electrically conducting fluids as described by the equations of magnetohydrodynamics (MHD). The MHD model is frequently used to model fusion and astrophysical plasmas. In this project, we aim to design and implement structure-preserving (SP) methods for MHD. 

SP methods aim to capture key properties of continuum models at the discrete level (such as conservation, symmetries, asymptotic limits, and maximum principles), and often improve robustness and accuracy in long-term simulations. 

For the MHD model, we aim to maintain divergence-free magnetic fields, conservation properties, and retain accuracy and efficiency when the system is close to equilibrium, characterized by low flow and Alfvenic Mach numbers. 

We will design and analyze a method based on discontinuous Galerkin phase-space discretization. 

The student will implement numerical methods in a high-level programming language (e.g., Python or Julia), analyze results, be introduced to computational MHD, and have the opportunity to interact with postdocs and lab staff.

Disciplines: Computational Mathematics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

RNA segmented viruses have independent genomic segments, or casettes, that are readily swapped when assembling viral particles. Thus, when a host is co-infected with two or more circulating viral strains, it is possible for a novel virus to emerge from the reassortment of casettes. This mode of evolution is more than just academic: for influenza A, the best characterized RNA segmented virus, reassortment has facilitated the formation of pandemic strains in 1957, 1968 and 2009.

Genome segmentation has important implications for viral gene expression control and RNA assembly into nascent virions. It also creates the potential for reassortment: the exchange of intact gene segments between viruses that co-infect the same cell. Reassortment is different from recombination since it allows many distinct genotypes to emerge from a single co-infected cell. Not only does segmentation enhance genetic diversification but it plays a unique role in the evolutionary history of segmented viruses due to the rare occasions when a reassortant is successful at a population scale. A striking example from the Bunyaviridae family of the emergence of a novel virus through reassortment is that of Ngari virus. There are six groups of segmented RNA viruses that infect vertebrates. Of relevance to this project are the Bunyavirales, given that of seven epidemic-prone diseases prioritised by the WHO 2018 R&D Blueprint as public health emergencies with an urgent need for accelerated research, three are Bunyaviruses: Lassa, Rift Valley and Crimean-Congo hemorrhagic fevers.

The intern will assist with the development of a mathematical model to quantify the reassortment probability in Bunyavirus co-infection. The intern will also assist with statistical and multi-variate analysis of potential data sets (published in the literature), including but not limited to principal component analysis, fitting, and Bayesian inference. The intern will join a well-rounded group of scientists in various disciplines like mathematical biology, statistical epidemiology, theoretical immunology, and theoretical virology, join office meetings, present results, and may assist with writing a research paper.

The intern should have experience with the study of differential equations, branching processes, Markov birth and death processes, algorithm development, computational methods (Python), Mathematica (or equivalent), multi-variate statistical analysis, as well as an interest in virology and immunology.

Disciplines: Applied Mathematics, Mathematical Biology, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM or virtual

U.S. Citizenship is a requirement for this internship

Project Description:

The research project Auto Hex-Dominant Meshing combines the merits of state-of-the-art geometric reasoning and artificial intelligence (AI) techniques in Sandia’s well-known geometry and meshing toolkit Cubit®.   The hex-dominant meshing has the potential to automate discretizing contiguous 3D domain similar to tet meshing and still provide accurate analysis results similar to hex meshing.  The contiguous 3D domains in many industries are represented using the boundary representation (B-Rep) in many CAD formats.  As B-Rep captures only the skin of a 3D domain, we first extract a skeletal representation called Chordal Axis Transform (CAT).  The CAT reduces the 3D shape into a 2D shape with an associated thickness field information.   The CAT representation assists in identifying thin-wall regions, T-junctions, holes, extremities, etc.  The CAT is further reduced to a weighted graph representation and a few graph algorithms are used to automate geometry decomposition for hex-dominant meshing.   As this is a new approach, further research is required to enhance geometric reasoning and artificial intelligence techniques on the graphs to improve efficiencies and robustness of hex-dominant meshing.  Students will be exposed to Cubit®’s geometry, meshing, and AI capabilities, programming, and potentially writing research finding for an international conference.        

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, and Geometric Analysis, Algebra or Number Theory

Hosting Site:

Sandia National Laboratories (SNL)

Internship location: Virtual

Project Description:

Modeling dynamics of a disease propagating in a population is important to predict health care needs and effectiveness of interventions and mitigation strategies.   Standard models of epidemics rely either on mean field approximations that abstract away real-world complexities or agent-based simulations requiring extensive fitting to real-world spatial correlations that are often difficult to measure accurately.  This project seeks to develop models that bridge the gap between mean field and agent-based models.

The starting point of this project is to view the epidemic from the point of view of a single individual.  For that individual, we want to describe statistically the time to infection by modeling the hazard of getting infected.  Such a model involves knowledge of the dynamic of the disease in the general population, but it also will account for individual characteristics such as connectedness of the individual to the population and its susceptibility to infection.

Our project will seek to aggregate an ensemble of “single individual hazard models” to build a stochastic description of disease evolution in a population, using the language of point processes.  Benefits of this new formalism is a better accounting of individual heterogeneity and opportunities to leverage ideas and analysis developed for point processes.   Such a formalism is particularly useful to model co-infections.

A mathematical description of the aggregation involves modeling dependencies between random processes, approximating the distribution of sums of dependent variables, and applying ideas from point processes to disease modeling.  As such, the ideal intern should have a background in applied mathematics, statistics or probability, some experience modeling stochastic processes (and point processes in particular), and an interest in disease modeling.

The successful candidate will have the opportunity to participate with a team of disease modelers at the Los Alamos National Laboratory, will be encouraged to give talks and poster presentation of the project, and expected to contribute to a manuscript to be published in a peer reviewed journal.

Disciplines: Applied Mathematics, Mathematical Biology, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

Project Description:

Under the direction of the mentor, the participant will learn about optimization under uncertainty and its application to the bulk electricity system both mitigate and adapt to climate change. They will also have the opportunity to learn about deploying optimization algorithms on DOE high-performance computing systems. Under the mentor’s guidance, the participant will develop mathematically sound approaches for transmission and capacity expansion as applied to the bulk electricity system to enhance the economics, reliability, resilience, and security of real-time grid operations as power systems both decarbonize and adapt to climate change.  A final report and presentation will be delivered on the project at the end of the appointment.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

In science and engineering domains, we often turn to mechanistic computer simulation models to predict the behavior of natural systems, such as fluid dynamics, molecular dynamics, or particle interactions in a collider. These simulations are often based on differential equations and provide critical information about the underlying governing equations but often are computationally expensive. To address this limitation, machine learning based approach is neural partial differential equation (NPDE), which seeks to emulate a complex computer model with a simpler partial differential equation (PDE) whose governing equation or “right hand side” is learned from data in the form of a neural network. However, existing work typically do not address the uncertainty associated with the neural-based emulation model arising from limited training data and function approximation errors.

In this project, we will first explore the potential of NPDEs to describe various physical systems, such as those governed by the heat or wave equations, or more complex systems like reaction-diffusion processes describing spatially distributed chemical reactions, or phase separation in self-assembling nanomaterials. Subsequently, we will explore ways to quantify the neural network uncertainty via a Bayesian deep learning framework. To recover useful physical equations from limited simulation data, we will also explore the reduction of overparameterized neural network weight spaces through linear subspace methods and Bayesian inference on this reduced parameter space.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Brookhaven National Laboratory (BNL)

Internship location: Upton, NY

Project Description:

Graph Neural Networks (GNN) have a wide range of applications in social and physical sciences. Improving the execution efficiency of these networks is notoriously difficult because the performance relies on the sparsity and connectivity pattern of these networks and the mapping to the underlying system. 

This project aims at analyzing the performance of these GNNs in HPC environments. We also aim to apply transformations to the GNN model to improve their representations and execution efficiency, considering application and system constraints.

Students will gain knowledge of executing these in leadership class HPC environments and performance analysis and tuning techniques, algorithmically and through transformation to the model and data representation.

Student Requirements: Basic knowledge of deep learning frameworks, such as Tensorflow or PyTorch. Familiarity with performance analysis techniques is desirable but not required.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA or virtual

Project Description:

This project aims to develop scalable randomized algorithm for data collection and processing in Bayesian inverse problems. Inverse problems use experimental data to infer parameters governing physical models (e.g., initial or boundary conditions of a partial differential equation). For scenarios where data collection is time-consuming or expensive, optimal experimental design (OED) for inverse problems seeks to determine experimental conditions for data acquisition that minimize uncertainty} (or maximize information gain) in parameters, predictions, or other quantities of interest---subject to budgetary constraints. We will explore the use of Randomization to mitigate the computational challenges posed by the OED problems.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL or virtual

Project Description:

This project will explore efficient solutions methods for solving a Bayesian inverse problem (i.e. recovering model parameters from observations) in systems modeled by stochastic PDEs with high-dimensional input data. The solution methods will rely on surrogate models of the PDE solver - a cheap-to-evaluate function that maps a sample of the stochastic input to the dependent variable in the PDE. In order to efficiently construct a surrogate with limited data, we leverage several key building blocks from state-of-the-art deep learning that encode desirable inductive biases into the model. Additionally, we will also explore sampling methods such as Normalizing flows to sample from the posterior.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL or virtual

U.S. Citizenship is a requirement for this internship

Project Description:

Object detection algorithms have broad applications and this project is focusing on the application of detecting objects sitting on the ground surface amongst a cluttered background. The images have low signal-to-noise ratios and our aim is to minimize the false negatives. The datasets for this project are collected using longwave IR and polarized longwave IR.

The MGI intern will explore approaches to combine traditional computer vision detection approaches with machine learning for optimal performance in a cluttered background. Traditional computer vision approaches behave in a predictable way on out-of-domain datasets and this project aims to incorporate computer vision features into a machine learning object detection algorithm to result in a more robust detection algorithm.

The intern will participate in a multi-disciplinary research group, participate in laboratory meetings, and have the opportunity to showcase results.  The intern should have some basic experience with algorithm development and machine learning (pytorch). Experience with computer vision is beneficial, but not required.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

U.S. Army Corps of Engineers, Cold Regions Research and Engineering Laboratory (CRREL)

Internship location: Hanover, NH

U.S. Citizenship is a requirement for this internship

Project Description:

Understanding the operating conditions that lead to high or low probability of detection of objects using a given sensor is important in the development of a sensor decision aide tool. This project will focus on developing a machine learning algorithm to predict likely sensor performance given a set of operating conditions. This will require developing a reliable metric for probability of detection that is dependent on the dataset of interest and the sensor being used. This project will consider datasets collected with Longwave IR and polarized Longwave IR cameras with images of objects placed on the ground surface in a cluttered environment.

The MSGI intern will explore appropriate metrics to represent probability of detection, which may potentially make use of in-house developed algorithms. The intern will use these metrics to develop a machine learning algorithm that will take the input of the operating conditions (e.g., meteorological data) and output a probability of detection. There will be a model developed for each sensor type considered and the result of the project will lead into the development of a reliability measure for using a sensor in a given environment for object detection.

The intern will participate in a multi-disciplinary research group, participate in laboratory meetings, and have the opportunity to showcase results. The intern should have some basic experience with algorithm development, machine learning (pytorch), and data science.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

U.S. Army Corps of Engineers, Cold Regions Research and Engineering Laboratory (CRREL)

Internship location: Hanover, NH

Project Description:

In the current era of noisy intermediate scale quantum (NISQ) computing devices, the control pulses for realizing logical gates frequently need to be re-calibrated due to environmental noise and drift of system parameters. The dynamics of a quantum device can be modeled by Schroedinger’s or Lindblad’s differential equations, based on parameterized Hamiltonian and decoherence models in which the parameters, for example, are transition frequencies and decoherence times. Because of the probabilistic nature of the system parameters, they need to be represented by distributions rather than values. These distributions can then be used in a risk-neutral optimal control methodology to design control pulses for realizing logical gates that are more resilient to noise, and therefore require less frequent re-calibration.

In this project, the student will learn about probabilistic characterization of superconducting quantum devices and risk-neutral optimal control. The student will also implement a risk-neutral algorithm within our open source C++ code Quandary. As a stretch goal, the student will learn how to apply their control pulses on the Quantum Device Integration Testbed (QuDIT) at Lawrence Livermore National Laboratory.

Disciplines: Computational Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Livermore National Laboratory (LLNL)

Internship location: Livermore, CA

Project Description:

Disciplines: Applied Mathematics

Hosting Site:

Pacific Northwest National Laboratory (PNNL)

Internship location: Richland, WA or Virtual

Project Description:

Particle accelerators are a central tool for scientific discovery in various fields. From unraveling the governing laws of fundamental particles to advancing medical treatments and understanding the universe's origins. Ensuring the accuracy and reliability of these experiments is of critical importance and requires the development of robust beam control methodologies. Predicting accurate beam positions in particle accelerators via first-principles models is crucial for achieving optimal performance and safety in the physical experiment, potentially improving scientific outcomes. However, uncertainties in model parameters can compromise the reliability and robustness of beam position predictions. In this research project, we propose to address this challenge by applying worst-case optimization under uncertainty techniques to guarantee robustness of beam position predictions using Bayesian neural networks (BNNs). The project aims to develop an optimization framework that minimizes the worst-case uncertainty impact on the beam position predictions, ensuring reliable and robust control recommendations to experiment operators given uncertainty in the imperfect data-driven models induced by insufficient data. To achieve this, we will first build a BNN model to map applied current at alignment magnets to the particle beam position, wherein model-form uncertainties are captured by inferred probability distributions over the weights. This allows BNNs to capture and propagate uncertainty throughout the network's computations. Next, we will formulate an optimization problem that leverages worst-case uncertainty optimization principles to minimize the worst-case impact of uncertainties in the BNN model on the beam position predictions. By considering BNN uncertainties as bounded uncertainties and employing robust optimization techniques, we will identify optimal model parameter values that minimize the worst-case uncertainty impact, ensuring reliable performance even under our imperfect models. The proposed framework will be evaluated and validated using particle accelerator beam position data.

The student intern on this project will gain practical experience in cutting-edge machine learning topics (i.e., Bayesian neural networks) and applied mathematics (i.e., optimization under uncertainty, uncertainty quantification). In addition, the intern will also gain a practical understanding of the challenges and “art” involved in applying optimization under uncertainty and machine learning techniques to real-world applications, specifically in the field of particle accelerator control. Given the student’s specific interests, we can determine primary areas of focus for the scope of their internship.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Brookhaven National Laboratory (BNL)

Internship location: Upton, NY or virtual

Project Description:

With more and more evidence demonstrating the success of large-scale foundation models (deep learning neural networks trained with massive datasets) in many science and engineering applications, more efficient and accurate complex system modeling can be achieved leveraging these advancements to enable optimal decision making to alter system behavior towards specific operational objectives. The recent GraphCast AI model has shown superior prediction performance compared to the existing state-of-the-arts for global weather forecasting, for which previous models are computationally expensive to simulate and whose dynamics are substantially uncertain. Such large AI surrogate models open a door for complex spatio-temporal dynamic modeling in a more computationally tractable manner.

Disciplines: Applied Mathematics, Computational Mathematics, Mathematical Biology , and Probability and Statistics

Hosting Site:

Brookhaven National Laboratory (BNL)

Internship location: Upton, NY or virtual

U.S. Citizenship is a requirement for this internship

Project Description:

This is to develop a hybrid-AI based sensor fusion framework for auto-navigation of unmanned aircraft systems (UAS), which incorporates physics and human cognitive logic into AI to make the AI more predictable, explainable, and robust. Hybrid-AI is one of the cutting-edge technologies initiated very recently to address the unpredictable behavior of AI and training bias, which are just emerging as its limitations. Decreased dependency upon the quality of the training dataset, which constrains performance, will result in increased hybrid-AI accuracy and reliability if AI generated features based either on human cognitive logic or via fused physics-based approaches can be effectively controlled, especially for unseen scenarios.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

U.S. Army Corps of Engineers, Cold Regions Research and Engineering Laboratory (CRREL)

Internship location: Hanover, NH

Project Description:

State-of-the-art fluid separation technologies, such as in gas purification, water desalination and chemical processing, involve flows of fluid mixtures across nanoporous graphene and graphene oxide membranes. Fluid dynamics at the nanoscale is predominantly governed by thermal fluctuations and Knudsen effusion, where the classical Navier-Stokes equations are not valid, and one has to rely on a molecular description.

Our group has developed numerical methods for simulating continuum fluctuating hydrodynamics (FHD) for fluids at the nanoscale by incorporating stochastic fluxes that correctly account for intrinsic thermal fluctuations. Furthermore, we also have expertise in using Direct Simulation Monte Carlo (DSMC) methods for a high-fidelity, but computationally expensive, molecular representation of the nanoscale fluid.

We propose implementing an adaptive algorithm refinement (AAR) hybrid numerical method to simulate gas permeation across these membranes. Using this approach, the nanoscale fluid dynamics will have a high-fidelity DSMC representation in the region near the membranes, whereas a continuum FHD will be implemented far from the membrane for enhanced computational performance.

In this project, we will collaborate to develop and implement numerical methods to couple DSMC and continuum FHD representation of nanoscale fluid dynamics using AAR. This project will involve collaboration with a team of applied mathematicians and computational physicists in the Center for Computational Sciences and Engineering at Lawrence Berkeley National Laboratory.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA

Project Description:

We propose to develop structure-preserving ML-based surrogates for singularly-perturbed dynamical systems. Conventional solvers for singularly-perturbed systems, such as those arising from the semi-discretization of a kinetic PDE, are generally too expensive for deployment in an inner-loop of an optimizer.  Reduced-order and ML-based models are typically used as surrogates for these solvers, but they often lack the robustness necessary to accurately model multiple timescales and resolve the limiting dynamics. Our approach involves leveraging structure-preserving properties in ML-based surrogates as a novel strategy to address these challenges. We intend to embed specific mathematical or physical structures robustly to capture the asymptotic behavior associated with the singular limit. Our ML-based methodology will employ key mathematical tools such as numerical linear algebra, advanced time integrators, perturbation methods, and geometric singular perturbation theory. The focus of our applications will be on kinetic models relevant to fusion devices and accelerators, aligning closely with several DOE Office of Science programs.

Disciplines: Analysis, Applied Mathematics, and Computational Mathematics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM or virtual

Project Description:

Generative machine learning models, including variational auto-encoders (VAE), normalizing flows (NF), generative adversarial networks (GANs), diffusion models, have dramatically improved the quality and realism of generated content, whether it's images, text, or audio. In science and engineering, generative models can be used as powerful tools for probability density estimation or high-dimensional sampling that critical capabilities in uncertainty quantification (UQ), e.g., Bayesian inference for parameter estimation. Studies on generative models for image/audio synthesis focus on improving the quality of individual sample, which often make the generative models complicated and difficult to train. On the other hand, UQ tasks usually focus on accurate approximation of statistics of interest without worrying about the quality of any individual sample, so direct application of existing generative models to UQ tasks may lead to inaccurate approximation or unstable training process. To alleviate those challenges, this project will develop several new generative models for various UQ tasks, including pseudo-reversible conditional normalizing flow model and its convergence analysis, diffusion-model-assisted supervised learning of generative models, and a score-based nonlinear filter for recursive Bayesian inference. We will demonstrate the effectiveness of those methods in various UQ tasks including density estimation, learning stochastic dynamical systems, and data assimilation problems.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN or virtual

Project Description:

This project spans multiple disciplines: applied mathematics, chemistry, physics, and software development. Students will be supervised by field experts and only need basic coding skills and linear algebra knowledge. We will develop a fast, mixed numerical precision, electronic structure solver that combines cutting-edge high-performance computing with computational science. In standard approaches to quantum chemistry, such as Density Functional Theory, several computational challenges to calculating the electronic structure of molecular systems are encountered. Often, the main computational bottleneck is solving a non-linear Schrodinger type eigenvalue problem which requires calculating the so-called density matrix describing the electron density. Analytically, this density matrix is obtained by applying the Fermi-Dirac function to the Hamiltonian matrix, which practically translates into matrix diagonalization, an expensive and time-consuming operation. A robust and useful alternative is to instead approximate the Fermi-Dirac function with a Chebyshev polynomial expansion. Recently, a fast Chebyshev expansion method for GPUs was developed [1-2]. However, this algorithm is extremely ill-conditioned, with a condition number that scales exponentially in the polynomial expansion order. For small order expansions calculated in double-precision, this does not pose an issue. But in the case when large order expansions are desired, or a low numerical precision is used, accuracy of the resulting density matrix suffers substantially. In this project, we seek to develop and study strategies that mitigate this undesirable numerical behavior in order to obtain high accuracy. The GPU and AI hardware code will be written using CUDA and HIP (for Nvidia and AMD devices, respectively) or using the hardware agnostic MAGMA library [3]. We will develop and test this on LANL’s Darwin compute cluster which has Nvidia A100 GPUs and AMD MI250X GPUs. It may also be necessary to utilize vendor libraries such as Nvidia’s cuBLAS or AMD’s rocBLAS for linear algebra operations.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

Project Description:

The Limited-memory BFGS (L-BFGS) algorithm proposed by Broyden–Fletcher–Goldfarb–Shanno is a very efficient optimization algorithm. It makes use of the inverse Hessian matrix associated with the degrees of freedom to optimize, to accelerate the search for a local minima or maxima. It has been used to optimize the geometry of molecules using the forces obtained from the quantum electronic structure surrounding the atoms. One issue in this context though is that function evaluations and their derivatives can be noisy and lead to the algorithm failures when getting close to the energy minimum. In this project, the student will develop a simple L-BFGS code, couple it to an existing electronic structure library that will provide function evaluations and their derivatives, in order to explore optimal strategies for memory-length and line minimizations in the presence of noise.    

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

The interaction between two rigid bodies can be generally described by six degrees of freedom when taking their orientations into account. This is specifically the case for two rigid molecules, identical or not. In this project, the student will (i) explore how the potential of interaction between two molecules looks like using a recently proposed compact, symmetric representation of that space, using quaternions (ii) investigate efficient numerical approaches to represent such a function, possibly using machine learning techniques.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

The advent of high-performance computing capabilities, accompanied with highly optimized algorithms, has presented scientific computing with the formidable task of having to handle vast volumes of generated data. This trend has received a substantial impetus from the Exascale Computing Projects (ECP). In the field of Quantum Molecular Dynamics (QMD), the ECP project has enabled the simulation of considerably larger systems over significantly larger timescales. The challenge now revolves around analyzing the generated data to extract crucial information, including, for instance, the identification of chemical reactions. When a reaction happens, electrons get reorganized across the molecule, which translates into an increase of the electronic current flowing through the chemical bonds. In this project we will use the Liouville-von Newmann equation to compute the currents between bods and detect reactions by monitoring the sudden changes in current flows through the course of a QMD simulation. We will use the LATTE code [1] to perform QMD simulations. LATTE will be compiled using BML[2] and PROGRESS[3]  libraries which enables the use of GPUs such as the new A100 Nvidia GPUs recently incorporated on LANL’s Darwin HPC cluster. The student will learn to develop a scientific code using best software practices, including version control method and regression testing. We will use this code to produce simulations and visualize chemical reactions. This project can lead to a fast publication that could be written after the summer appointment. Some previous experience with HPC and Fortran programming is recommended, as well as a fair linear algebra math background.

1- Bock, N., M. J. Cawkwell, J. D. Coe, A. Krishnapriyan, M. P. Kroonblawd, A. Lang, C. Liu, et al. 2008. “LATTE.” https://github.com/lanl/LATTE.

2- Bock, Nicolas, Christian F. A. Negre, Susan M. Mniszewski, Jamaludin Mohd-Yusof, Bálint Aradi, Jean-Luc Fattebert, Daniel Osei-Kuffuor, Timothy C. Germann, and Anders M. N. Niklasson. 2018. “The Basic Matrix Library (BML) for Quantum Chemistry.” The Journal of Supercomputing 74 (11): 6201–19.

3- openhub.net. n.d. “The Qmd-Progress Open Source Project on Open Hub.” Accessed September 2, 2019. https://www.openhub.net/p/qmd-progress.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

Project Description:

Surrogate models are increasingly used in calibrating complex and expensive epidemic simulations. Such surrogate models are used to estimate unknown parameters in the simulation model when only a limited number of simulation runs are possible. Surrogate models are generally statistical or machine learning models that aim to learn the input-output relationships from limited simulation outputs. There exist a number of surrogate techniques, some focus on a specific set of aspects of the simulation but all are used as faster alternatives to running the actual expensive simulations the many times needed for characterizing model parameter spaces.

In this project, we will investigate an AI-based surrogate technique called neural ordinary differential equations (NODE) and apply it to an epidemic simulation setting. Since typical compartmental epidemic models can be expressed as a set of ordinary differential equations (ODEs), NODEs map naturally to encompass the underlying epidemic dynamics. Having an efficient surrogate model is particularly useful for large simulation models, such as city-scale agent-based models (ABMs) that our research group at Argonne has developed. In particular, the CityCOVID ABM models all of the residents of Chicago (2.7 million individuals) going to and from specific geographic locations (1.2 million locations) on an hourly basis. For this project, we will make use of a simpler (yet highly detailed) mechanistic epidemic model to generate synthetic disease trajectories for which the NODE surrogate will be trained. Later, we will apply the same technique to the more expensive CityCOVID model. This will require the tuning of the NODE surrogate structure and learning parameters (so-called hyperparameters) to develop accurate time series emulation of the model’s epidemic trajectories. Upon successful development of the NODE surrogate for the model, we will be able to apply it to develop fast model calibration approaches that can make use of the computational efficiency of the NODE surrogate and its ability to provide derivatives.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Virtual

Project Description:

Mesh generation has always been a time-consuming and error-prone process. This is especially true in computational sciences, where meshes must be generated for three-dimensional geometries of various levels of complexity. Inspired by past program synthesis work in constructive geometry, we will be using concepts from programming languages and formal logic to facilitate rule based synthesis of automated mesh generation algorithms.

Some of the goals of the project are:

* Formalize the mesh specifications and geometric characteristics of a mesh, which includes element size, element shape, and element orientation.

* Formalize geometric transformations of the mesh, and mesh quality enhancement techniques used by most production level meshing software.

* Study state-of-the-art mesh generation techniques like advanced front methods, Delaunay triangulation, and develop a refinement rule-based synthesis of the corresponding algorithms.

* The refinement method used to go from specification to code, proofs of correctness, and assertions related to mesh quality will be formalized in a state-of-the-art theorem prover like Coq, Lean4, Isabelle/HOL.

This project will be investigating the intersection between automated proof generation, numerical analysis, and computational geometry.

Disciplines: Computational Mathematics, and Geometry

Hosting Site:

Lawrence Livermore National Laboratory (LLNL)

Internship location: Virtual

Project Description:

This project would involve developing and evaluating robustness and uncertainty quantification methods for large language models (LLMs) and foundation models in scientific domains. The goal is to integrate probabilistic modeling techniques (e.g. conformal statistics, kernel attention) and advanced training procedures to enhance the reliability and controllability of these models in tasks related to climate modeling, chemistry, or cosmology. Furthermore, this would involve exploring adversarial robustness, uncertainty quantification, and fine-tuning strategies to mitigate inherent vulnerabilities and sensitivities in these models, contributing to the emerging field of robust and reliable AI in scientific research.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL or virtual

Project Description:

This project aims to leverage a novel unified Bayesian framework for network architecture search and ensembling developed recently in our group to build physics-preserved neural networks for coupled PDE-driven problems such as nuclear fusion and fission. By integrating sparse priors and innovative pruning strategies, the framework not only discovers optimal architectures but also quantifies uncertainty at the neural architecture level. The focus will be on incorporating physics information from differential equations directly into the neural architecture through such an approach, enhancing the model's performance and reliability.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL or virtual

Project Description:

Social attachments play a fundamental role in most, if not all, levels of human interaction. Secure attachments promote resilience, where insecure or disrupted attachments increase the risk of multiple neuro-psychiatric conditions. Despite its clinical significance, social attachment has been difficult to study because traditional model systems like mice do not display long-term social attachment behaviors. Prairie voles strongly engage in social behavior, and further allow to study genetic effects on the social repertoire, including mutations implicated e.g. in autism spectrum disorder (ASD).

Understanding the relationship between the brain and behavior requires an integration of neurobiology and ethology in the study of naturalistic behaviors. Progress toward this goal requires the development of tools and analytical frameworks capable of measuring pose and describing pose dynamics. Recent advances in computational neuroethology have enabled high-fidelity, high-throughput tracking of animal pose as well as the detection of recurrent behavior modules. However, these existing methods are geared toward the study of individual, rather than social, behaviors.

Quantification of the social interactions would permit the identification of shared structures in behavior syllables and neural recording data, yielding hypotheses about the neural mechanisms that underlie behavior selection.

In this project we aim (i) to develop advanced machine learning methods to extract and track animal postures from (single or multi-view) video recordings of socially interacting animals and (ii) to develop novel algorithms to identify socially interacting motifs.

In particular, we will design and train a attention based neuronal networks that track animal postures and skeletons over time by extending current deep object transformer neuronal networks which provide state of the art video object segmentation. A focus will be on extracting postures of interacting animals that frequently occlude each other and apply this to our data sets of socially interacting prairie voles.

We will further develop machine learning methods based on inference of switching dynamical systems to detect and extract behavioral motifs form the tracked data. In particular. we will employ hierarchical Dirichlet process hidden Markov auto-regressive modeling and extend existing approaches to the situation of multiple interacting agents.

The tools developed here will have larger range of applications in the analysis of interaction dynamics in complex systems science.

Disciplines: Applied Mathematics, Computational Mathematics, Mathematical Biology, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Virtual

Project Description:

Although increasingly general models like GPT-4 have made remarkable strides to wide acclaim, subsequent research have revealed surprising limitations in their ability to perform basic logic, planning, and reasoning. This summer experience will provide a curious student the opportunity to explore neurosymbolic AI and its application to the design and control of energy systems.

The student will be mentored by a team of NREL AI scientists to conduct and analyze approaches to AI that combine modern, state of the art neural network-based methods with classical symbolic approaches. The research will involve integrating AI techniques, such as reinforcement learning, with classical mathematical optimization approaches and implementations, where symbolic and sub-symbolic representations are freely intertwined, and learning is ubiquitous. Approaches inspired by cognitive science are especially welcome.

Successful students will have creative problem-solving skills and interest in cross-disciplinary collaboration. Familiarity with mathematical and statistical foundations of both neural-based AI (statistical learning) and symbolic AI (reasoning) is a plus.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

Machine learning methods for time series play an important role across a spectrum of scientific applications, including renewable energy forecasting and studying the dynamics of complex nonlinear systems. The success of deep learning and increasing availability of large datasets has created interest in global forecasting models. These models are pretrained on a collection of related time series and can be fine-tuned after pretraining on target time series.

Current research challenges for global time series modeling are as follows: 1) related time series can have dynamic ranges that vary widely, even within a single dataset, 2) time series may be continuous, discrete, or mixed, and each may be distributed in arbitrarily complex ways. At the same time, probabilistic modeling over future scenarios remains difficult, yet is critical for many scientific applications.

This summer project will engage a student passionate about deep learning, applied math/probability/stats, and energy/climate applications to design a new probabilistic approach for global time series forecasting. Learning opportunities can include (but are not limited to): 1) implementing and evaluating existing advanced probabilistic deep learning methods for global time series forecasting (e.g., C2FAR (Bergsma et al., 2022)), 2) integration of these approaches into an existing benchmark for generalizable building energy forecasting (BuildingsBench: Emami et al., 2023), and 3) propose a novel method to be written up in a report (possibly for publication).

Familiarity with PyTorch is a prerequisite for success. The student will be paired with an NREL machine learning scientist as a mentor to guide them along the way.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

Aerosols are small particles (<~10 microns in diameter) that are suspended in a gas, like the Earth’s atmosphere. These particles are central to understanding the water cycle and transport of nutrients within the Earth System. A complete understanding cannot be provided without ground-based and vertically resolved observations, particularly for aerosol-cloud interactions. The goal of this study is to analyze diverse and large datasets collected by the U.S. DOE Atmospheric Radiation Measurement (ARM) mobile facility during campaigns that we have deployed instrumentation to and are actively involved in with funding from the Atmospheric System Research (ASR). Two examples include the Surface Atmosphere Integrated Field Laboratory (SAIL) campaign in Colorado focused on aerosol impacts on mountain hydrology and the Eastern Pacific Cloud Aerosol Precipitation Experiment (EPCAPE) in California focused on aerosol-cloud interactions. Data includes but are not limited to those collected by the ARM, including the Aerosol Observing System (AOS) and the tethered balloon system (TBS) as well as those deployed by LANL with complex 2D and 3D datasets of size-resolved and single particle chemical composition, such as those collected using online and high-time resolution aerosol mass spectrometry. Specific aims are directed at answering key aerosol process science objectives related to identifying different aerosol regimes, the processes controlling their lifecycles, quantifying impacts on the radiative budget, and the sensitivity of cloud phase and precipitation to cloud condensation nuclei (CCN) and ice-nucleating particle (INP) concentrations.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM or virtual

Project Description:

Adaptive signal timing at signalized intersections crucially relies on precise vehicle arrival forecasting. In this project, we present a cutting-edge Physics-Informed Language Model (PILM) specifically formulated to improve these predictions. The PILM harnesses observable patterns from traffic data across upstream and downstream detectors, fusing them with the immutable laws of vehicle conservation and sophisticated language processing algorithms that decode both temporal and spatial aspects of traffic sequences. A novel tokenization approach is proposed to tokenize the vehicle arrivals by time along the prediction horizon. The model dynamically incorporates signal timing and queuing theory to account for the complex interactions at intersections. Our model adopts a data-oriented paradigm, allowing it to internalize and anticipate spatiotemporal vehicular arrival patterns, while its physics-aware framework guarantees conformity with tangible traffic movements. Through extensive data modeling and simulations, we demonstrate that our model outperforms existing prediction methods in accuracy and robustness, especially during peak traffic conditions and incidents that disrupt normal flow patterns. The PILM’s predictive insights are critical for intelligent traffic management systems, facilitating improved adaptive traffic signal timing, reducing congestion, and enhancing road safety.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

This project is to explore different graph neural networks and message passing algorithms for functional annotation of protein residues from protein three-dimensional (3D) structures. Proteins are polymers of length N built from 20 different residue types, where N ranges from tens to thousands of residues. Different arrangement of the 20 residues along the protein sequence affects the 3D structure and function. Engineering a protein for specific function requires altering residue types at certain positions along the sequence. Identifying the positions and residue types to change is a combinatorial problem where the search space is prohibitively large with 20N possible choices. A solution to address this challenge is to accurately predict function at the residue level so that the search space is narrowed down to specific residue targets that contribute the highest to the function of interest. Graph neural networks have been used to represent protein 3D structures, where atoms of residues are nodes and connectivity between atoms are edges. However, to use graphs for function prediction, it is necessary for graphs to capture relevant physicochemical properties of residues and residue-residue interactions. Scientifically, it remains unclear what the relevant residue properties and interactions are, and mathematically, it is unclear which graph models are suitable for such representations. To this end, the student will develop and apply different graph neutral networks and message passing algorithms to study protein structure-function relationship at the residue level. The students will use a dataset comprised of human proteins involved in three biological functions as a benchmark. The goal is to understand how different graph neural networks and message passing algorithms affect resulting residue function prediction and identify key residue properties and interactions for the three functions of interest.

Learning objectives:

1) Learn the impact of different graph neural networks and message passing algorithms on functional annotation of protein residues by participating in model implementation and performance analysis

2) Learn the relationship between protein structure and function

3) Meet staff, present research findings, and practice scientific communication skills with broad audience

Disciplines: Applied Mathematics, and Mathematical Biology

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

Transformers are neural networks that use attention mechanisms to capture long-range dependencies and context in sequential data. They have revolutionized natural language processing, computer vision, and other domains that deal with complex and high-dimensional data. Power systems are also dynamic and nonlinear systems that require efficient and robust methods for analysis and control. This project will involve utilizing the Transformer architecture and adapting its applications and mathematical foundations for power system applications, such as short-term voltage stability assessment, renewable energy integration, and fault detection.

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, Probability and Statistics, and Topology

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

Large Language Models (LLMs) are powerful neural networks that can process natural language and generate meaningful text. However, how LLMs make the correlations between mathematical computations and the topological information in graph networks is still an open question. Graph networks can represent entities and their relationships as nodes and edges, respectively - widely used to model complex systems and phenomena in various clean energy domains. This project will involve learning about the LLMs and their applications, mathematical foundations, and graph algorithms. It will also explore the methods and prospects of integrating LLMs with graph networks, such as using LLMs to enhance node attributes, perform graph reasoning, and generate graph structures.

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, Probability and Statistics, and Topology

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

This project is to explore acceleration strategies for low-rank training of neural networks. From a mathematical perspective, training neural networks is a large-scale optimization problem, involving billions of degrees of freedom. Training of neural networks is usually performed in Pytorch or similar Python APIs. 

Low-rank compression and training strategies aim to find low-rank representations of a neural network at hand to enable memory and run-time efficient training procedures. Here the neural networks are decomposed in an SVD-like fashion. Training, i.e. updating the parameters with a numerical optimizer, is performed in only on the factors of the low-rank matrix representation.

One key aspect of low-rank compression is the augmentation and truncation of the basis that is used to represent the low-rank network. Numerically this requires the repeated evaluation of linear algebra routines as QR decompositions and SVDs. Although many aspects of neural network training with Pytorch can be efficiently performed on a GPU, the implementations of QR and SVD algorithms are lacking in terms of wall-time efficiency. An efficient implementation of these linear algebra routines is an important aspect of low-rank training that determines the broad applicability of the method.

To this end, the student will assess the currently available standard implementations of linear algebra routines in Pytorch on modern general-purpose GPUs. The student will perform a broad runtime comparison for different low-rank neural network architectures and hardware configurations. After mapping the landscape of available methods, the student will implement the best-performing packages into the codebase for dynamical low-rank training.

The goal is to gain an understanding of high-performance neural network training at scale and contribute to the advancement of memory and wall-time efficient machine learning.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

Predicting alloy chemical properties given a set of material characteristics and environmental conditions like temperature is a challenging task due to noisy data, missing features, and multimodality.  An experimental approach to evaluate such alloy is expensive and highly inefficient. Generative AI can aid domain scientists in inverse-design of predicting feasible chemical properties under unseen or partially observed material characteristics and environmental conditions.

This project aims to explore and (i) implement transformer architecture (ii) learn techniques to tackle missing and noisy data samples, and (iii) explore data-driven architecture for multimodal features.

Student responsibilities: The student is expected to be comfortable with pytorch and python data analysis and visualization libraries like, numpy, pandas, matplotlib, scikit-learn. The student should have some foundational knowledge on deep learning. Having experience with transformer architecture or multimodal data is a plus, but not necessary.

Learning Objectives:

1. The student will learn basic concepts about different open-sourced transformer architectures
2. The student will be exposed to training deep learning models with missing and noisy data.
3. The student will familiarize with state-of-the-art machine learning and deep-learning models to research multimodal data.

Disciplines: Analysis, and Applied Mathematics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

In recent years, there have been several works investigating the use of quantized tensor networks (QTNs) to approximately solve nonlinear partial differential equations (PDEs) with significantly reduced computational cost. Example PDEs include the Navier-Stokes and the Vlasov equations; solving these equations using traditional numerical methods is often impractical or even infeasible due to extremely large computational costs. While the initial investigations appear quite promising, further work is needed for practical usage. For example, current works typically consider box-like geometries, though many problems of interest may be better suited for a cylindrical or spherical coordinate system. However, these coordinate systems require specialized methods to treat the singularity that appears in the grid. This project will involve investigating how these specialized methods can be translated into the QTN framework, and then developing QTN-specific algorithms that offer similar performance but with reduced cost. Students will have the opportunity to become familiar with both traditional numerical simulation algorithms and the growing field of quantized tensor network algorithms.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA or virtual

Project Description:

Disciplines: Combinatorics, and Probability and Statistics

Hosting Site:

Pacific Northwest National Laboratory (PNNL)

Internship location: Virtual

Project Description:

Obstructive sleep apnea (OSA) affects 24% of all Veterans or 1 in every 15 Americans and is associated with increased risk for developing cardiovascular and metabolic comorbidities such as heart disease, stroke, hypertension, and type 2 diabetes mellitus. A vast number of patients are already experiencing at least one of these comorbidities by the time they are diagnosed with OSA; suggesting that OSA is being diagnosed well after the onset. OSA shares many symptoms with other diagnoses such as depression, increasing the diagnosis gap. With the use of machine learning and natural language processing, as well as statistical modeling and deep learning, we can better understand and identify OSA and target patients for treatment closer to the onset, reducing the likelihood of patients developing comorbidities. Applicants will have the opportunity to participate with many clinicians and subject matter experts. Applicants will have the opportunity to use Perlmutter, one the largest supercomputers in the world. Applicants will have the opportunity to learn, explore and deploy large language models, predictive models and more on healthcare data.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA or virtual

Project Description:

Many scientific machine learning (SciML) seek to quickly approximate the solutions of partial differential equation (PDE) systems. These include physics-informed neural networks, convolutional neural networks, graph neural networks, attention-based Transformer models, and neural PDEs. Generative models of the Transformer type, in particular, have revolutionized natural language processing (such as ChatGPT), and by now several ""foundation"" models exist for scientific applications such as climate/weather, biomedicine, materials science, etc.

However, while these data-driven models may serve as useful predictors or proxies for numerical PDE simulation, it is not yet clear what they learn about physical systems or why. General results from the ""explainable AI"" (XAI) field have provided means to study how input features influence outputs (e.g., how the initial state is related to the predicted state); how individual neural network layers act on inputs to produce predictions; and even how Transformer layers learn to approximately perform iterative numerical algorithms such as first- or second-order optimization.

This project will involve applying XAI techniques to generative SciML models trained on linear and/or nonlinear PDE solution data. One objective could be to relate properties of the trained ML model (such as its accuracy and precision) back to properties of the underlying dynamical system, such as its eigenspectrum, stiffness, degree and type of nonlinearity, symmetry and conservation laws, etc. Another could be to study the error of the neural network's predictions as they relate to the amount and type of training data provided. The project will provide opportunities to train neural networks, analyze their mathematical structure and properties, as well as to use numerical PDE solvers and analyze the behavior of dynamical systems.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Brookhaven National Laboratory (BNL)

Internship location: Upton, NY or virtual

Project Description:

Large-scale simulation models are pervasive throughout science, used in climate, materials science, nuclear physics, quantum chemistry, etc. These simulations often require large amounts of computation time, scientific machine learning (SciML) methods have been developed to create AI-based ""surrogate"" models that make fast, approximate predictions of physical systems. These ML surrogates are trained to simulation output and learn to reproduce it under new input settings (initial conditions, boundary conditions, parameter values). However, once trained, the ML surrogate may not make further use of simulation data. Another approach, known as active learning, involves selecting new simulations to perform and then retraining the ML surrogate.

An alternative that has not yet been widely explored is whether the ML surrogate can be given access to a simulator that it can invoke on-demand, to generate limited amounts of new data, each time it makes predictions. The new simulation data would become additional inputs (predictors or covariates) to the trained surrogate, rather than a source of data to retrain the model. This would allow the ML model to generate, via physical simulation, additional information about the physical system that are expected to most improve its predictions. This project would explore whether additional ML-generated data of this form are useful in physical predictions, and how an ML model can learn a policy that controls which new simulation information is requested for a given prediction. The project would involve the opportunity to develop and train SciML surrogate models, integrate them with physical simulations, and analyze and validate their predictions.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Brookhaven National Laboratory (BNL)

Internship location: Upton, NY or virtual

U.S. Citizenship is a requirement for this internship

Project Description:

Researchers at ORNL use chemical vapor deposition (CVD) to produce TRi-structural ISOtropic particle (TRISO) fuel.  There are numerous variables in the CVD process that affect the uniformity of the TRISO fuel surface. This project will focus on developing statistical, machine learning, and reduced order models to describe the relationship between the variables in the CVD process and the uniformity of TRISO fuel and how we can use these models to optimize the fuel surface.

The student is required to have a strong background in mathematical/statistical/machine learning modeling;

The student will learn to apply mathematical and statistical modeling knowledge to an engineering process by studying the existing data and building models;

The student will learn new optimization methods by using the models as surrogates.

The student will help produce a code that will act as a library of mathematical/statistical/machine learning models to be applied to datasets to build surrogate models. The project will be summarized in a section of a technical report and should be given in presentations.

Events / opportunities happen inside of our directorate or division that the student will be able to participate in and learn from that aren't specifically related to this internship project include seminars, lunch and learns, and crosscut forums.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

As renewable energy systems, from mobility to human behavior models, become larger and more interconnected—with agents’ actions triggering a cascade of reactions nearly instantaneously—there is an increasingly urgent need for a method to simulate event-driven complex systems on high performance computing resources. Parallel discrete events simulations (PDES) are used to model a wide range of complex systems. NREL’s High Performance Algorithms and Complex Fluids (HPACF) Group is particularly interested in the use of PDES for solving large-scale event-driven renewable energy systems.  These systems include mobility and transport technology, the electrical grid, and models of material growth (e.g. crystals in chemical vapor deposition reactors). The objective of this research is to develop a scalable engine for PDES by investigating novel numerical methods and algorithms suitable for heterogeneous computing architectures, e.g., Graphics Processing Units (GPU), in high performance computing systems.

This project will build on ongoing work to accelerate solutions in PDES. The focus of this research will be on the use of GPU to solve discrete events systems. The student will build fundamental understanding of PDES algorithms and how these can be modified to target new computing architectures. Research will be tied to actual renewable energy technology research, including applications to material growth and mobility. This is a relatively open field with significant opportunities for future research and publication.

The intern can expect to build skills in the following areas:

1) High-performance computing.

2) Algorithm development for PDES.

3) Applications of PDES to renewable energy technology.

Interns in HPACF generally are encouraged to present internally and externally and submit results for peer review.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

Autonomous experiment design comes at a high computational cost. In this project the intern will investigate the mathematical equivalence or difference between experiment design strategies.

The goal is to write a paper providing the math of the equivalence or difference between strategies and then demonstrate using a real autonomous application.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

National Institute of Standards and Technology (NIST)

Internship location: Gaithersburg, MD or virtual

Project Description:

Resilience refers to the ability of a system to maintain certain degree of well-being in the face of disturbances. Complex systems, natural or engineered, are resilient if they possess feedback mechanisms of control that allow them to adapt after disturbances. Understanding the properties of those adaptation mechanisms is a key problem in many scientific areas in engineering, earth and social sciences.

Resilience in complex systems can be mathematically studied within the theory of random dynamical systems. Specifically, we are interested in the probabilistic response of high-dimensional noisy dynamical systems to exogenous perturbations. Here, randomness is included to account for the presence of uncertainty in initial conditions and/or model parameterizations. The goal is to study models of how systems can adapt dynamically in order to steer the probability distribution of its state away from states of low well-being, functionality or productivity.

In this project, we will model and analyze complex systems through a variate of mathematical tools. These include differential equations, probability, stochastic  processes and network theory. The student can expect to be exposed to and learn theoretical, applied and computational aspects of each of these subjects.

Specific areas of application can be decided according to the student’s background and interests. However, there are some subjects in which we already have ongoing projects: 1) Biology: resilience of photosynthesis pathways to temperature changes. 2) Ecology: resilience of trophic networks to changes in species variability or network topology. 3) Energy: resilience in electrical grids through improved power restoration protocols.

Disciplines: Applied Mathematics, Mathematical Biology, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

This project aims to investigate the use of physics-informed surrogate models for partial differential equations (PDEs) in computational fluid dynamics (CFD) applications. We will explore various strategies to incorporate adjoint information from PDE solvers in surrogate model training and develop optimization schemes that leverage adjoint information from surrogate models to accelerate PDE-constrained optimizations for CFD applications. In this project, we will learn about standard physics-informed surrogate modeling approaches for PDEs, basics of PDE solvers for CFD applications, and adjoint-based PDE-constrained optimization strategies. We will analyze the performance and efficiency of the developed surrogate-based optimization scheme using numerical analysis tools.

Disciplines: Analysis, Applied Mathematics, and Computational Mathematics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

Neural networks have complex connections, a variety of architectures,  and tunable parameters. The capacity of these networks to find non-trivial connections between inputs and outputs, or to accomplish complex tasks, is attributed to their complex structure.  This complexity delivers remarkable fidelity, but often at the expense of understanding. Moreover, the results are sometimes very fragile, often dependent on the richness of the training set. For certain and very common networks, one can frame the process of tuning parameters and of training as a dynamic learning problem of complex interacting agents. There is an existing framework for understanding systems of this sort, namely, statistical physics.

In this project we will apply ideas from equilibrium and non-equilibrium statistical physics to tune simple networks and endow them with some degree of explainability. The project combines theory, computation, and numerical experimentation. The two essential skills required to participate in this research project is probability (there is no requirement for a proficiency in measure theory), linear algebra, and some expertise in coding in python, Julia, or matlab. The research performed by the student will be reported eventually in peer-reviewed scientific journals.

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN or virtual

Project Description:

In this project, the student will explore methods to fuse a heterogeneous set of materials science data using Bayesian statistics. The student will implement specific prior distributions using kernel-based assumptions and perform sample-based inference on marginal distributions. The student will implement models using standard Python libraries and Python-based Markov chain Monte Carlo packages. Students will explore the effectiveness of various kernels and MCMC parameters, applying their developed tools to data from the Materials Project database.

Disciplines: Probability and Statistics

Hosting Site:

Brookhaven National Laboratory (BNL)

Internship location: Upton, NY or virtual

Project Description:

Crystal shapes play a pivotal role in determining the properties of energetic materials, impacting their performance and safety [1, 2]. Various theoretical methods, such as the attachment energy model and kinetic Monte Carlo simulations, exist for predicting crystal shapes [3, 4]. However, discerning whether the predicted shape is at equilibrium remains a challenge. This project aims to address this gap by obtaining the crystal shape of an energetic material, pentaerythritol tetranitrate (PETN), at equilibrium. The use of free energy methods, such as umbrella sampling (US), in molecular simulations is powerful for obtaining the equilibrium state of a system [5]. In this project, we will perform biased molecular dynamics (MD) simulations using the US method to determine the equilibrium shape of the crystal.

The student undertaking this internship will have the opportunity to develop essential skills in computational chemistry, including thermodynamics, classical mechanics, and statistical mechanics. The student will learn to perform molecular simulations on high-performance computers. These skills will not only be valuable for this project but will also lay the groundwork for future research endeavors in computational chemistry. The student is expected to have an interest in physics with coding skills in either Python, C++, or Fortran.

1- Handley, C. A., et al. "Understanding the shock and detonation response of high explosives at the continuum and meso scales." Applied Physics Reviews 5.1 (2018).

2- Perry, W. Lee, et al. "Relating microstructure, temperature, and chemistry to explosive ignition and shock sensitivity." Combustion and Flame 190 (2018): 171-176.

3- Ibrahim, S. Fatimah, et al. "Prediction of the mechanical deformation properties of organic crystals based upon their crystallographic structures: case studies of pentaerythritol and pentaerythritol tetranitrate." Pharmaceutical Research 39.12 (2022): 3063-3078.

4- Zepeda-Ruiz, Luis A., et al. "Size and habit evolution of PETN crystals—a lattice Monte Carlo study." Journal of crystal growth 291.2 (2006): 461-467.

5- Singh, Himanshu, et al. "Determination of equilibrium adsorbed morphologies of surfactants at metal-water interfaces using a modified umbrella sampling-based methodology." Journal of Chemical Theory and Computation 18.4 (2022): 2513-2520.

Disciplines: Analysis, Applied Mathematics, Biometrics and Biostatistics, Computational Mathematics, Geometry, Mathematics (General), Operations Research, Probability and Statistics, and Topology

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

Project Description:

Graph Neural Networks (GNNs) have gained significant attention owing to their ability to handle graph-structured data and the improvement in practical applications. However, many of these models prioritize high utility performance, such as accuracy, with a lack of privacy consideration, a significant concern in modern society where privacy attacks are rampant. In this project, we want to focus on privacy-preservation techniques. With guidance from a mentor, the student will help developing new defense mechanisms and design comparison study with the state of the art methods. The student will learn about privacy-preservation algorithms, graph neural networks numerical analysis, and writing/presentation skills.

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, Mathematics (General), and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

Standard computational fluid dynamics approaches have widely relied on solving the Navier-Stokes equations governing fluid flow. These approaches assume that the fluid can be treated as continuum, which can be ill-posed for complex fluid flows such as rarefied gases and hypersonic aeronautics. In such cases, it becomes necessary to revert to the governing equations of molecular gas dynamics which underpin the macroscopic behavior of the fluid. One such example, the Boltzmann equation, provides a statistical description of particle transport and collision which can seamlessly recover the hydrodynamic equations in the continuum limit while offering a more detailed description of fluid flow in the non-equilibrium regime. However, its high-dimensional nature drastically increases the associated computational cost of solving complex fluid flows.

The goal of this project is to design and apply robust and efficient numerical schemes for solving the Boltzmann equation, namely high-order unstructured finite element methods which retain structure-preserving properties such as positivity of probability density and the conservation of macroscopic quantities. The student’s research will use an established codebase on projects such as developing discretely-conservative particle velocity domain discretizations and collision models, nonlinear limiting and shock capturing methods, and adaptive mesh refinement techniques for the velocity domain. Further project ideas include the application of these techniques to complex three-dimensional flows including hypersonic re-entry vehicles and fundamental fluid flow phenomena such as transition to turbulence, developing data reduction methods and analysis for high-dimensional phase space representations of fluid flow instabilities, and any potential self-proposed project ideas by the student.

Students should have basic familiarity with high-order finite element methods, scientific computing, and computational fluid dynamics. Familiarity with molecular gas dynamics/Boltzmann equation is preferred but not necessary.

Disciplines: Computational Mathematics

Hosting Site:

Lawrence Livermore National Laboratory (LLNL)

Internship location: Livermore, CA or virtual

Project Description:

Our goal is to ignite research on quantum data analysis, quantum image processing, and quantum machine learning. We aim to develop innovative quantum data representations and quantum analysis methods applicable to different types of data by exploiting the recent progress in QIS, using quantum hardware and quantum simulators on high-performance computing (HPC) systems. We aim to optimize the new algorithms for current and future noisy-intermediate scale quantum (NISQ) devices, which are limited by low numbers of qubits with short decoherence times, and high gate error-rates, hindering the fidelity of quantum computations. The main goals of the project are to (1) develop and evaluate efficient quantum data representations and (2) develop implementations of quantum data analysis and quantum machine learning (ML) targeting essential data types (images, sequences, and time-series). We will explore hybrid quantum/classical algorithms and study their suitability and efficiency for specific analysis tasks.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA or virtual

Project Description:

A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. Commonly used in probability theory, statistics — particularly Bayesian statistics — and machine learning. The main advantages are its ability to use prior information (physical constraints) related to the data. This becomes very important when analyzing scientific data. This project aims to develop efficient PGM-based algorithms to tackle problems such as classification, regression, image segmentation, image denoising, feature tracking, data reduction, data fusion, etc. We use mainly Markov Random Fields and Conditional Random Fields, and some of our approaches combine these methods with deep learning algorithms to solve problems in different scientific fields such as chemistry, material sciences, and biosciences.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA or virtual

U.S. Citizenship is a requirement for this internship

Project Description:

Large quantities of CO₂ will be transported over a large distance from CO₂ capture plants to underground storage sites. There are growing safety concerns across communities that face the prospect of more CO₂ pipelines primarily due to accidental release of odorless toxic CO2 in the atmosphere caused by defective cracks on the pipe or pipe failure. It is necessary to have a validated risk assessment model to mainly predict the safety distance from the pipeline where accidental releases cannot cause unacceptable risk for the people.

The objective of this project is to extensively review existing dispersion models of carbon dioxide release in case of leak or rupture then select the best model to deepen our knowledge through extensive simulations of a combination of several scenarios.

These models are important to regulate the safety construction and operation of CO₂ pipelines through populated areas and remote communities.

Disciplines: Applied Mathematics

Hosting Site:

National Energy Technology Laboratory (NETL)

Internship location: Virtual

Project Description:

Iron and steelmaking are complex processes, with reacting multi-phase fluid flow and heat transfer occurring at extreme temperatures.  The need to decarbonize manufacturing, which accounts for as large of a share of carbon emissions in the US as electricity generation, has led to a renewed interest in understanding the physics of not just iron and steel manufacturing, but all manufacturing processes involving metals.  Because of the complex physics and extreme temperatures, simulation is critical to understanding these systems. These simulations then allow the impact of process changes, such as substituting hydrogen for fossil fuels in iron ore reduction, to be understood.  This, in turn accelerates re-engineering these processes to make them more sustainable. NREL’s High Performance Algorithms and Complex Fluids (HPACF) Group is host to multiple projects simulating these processes, using both commercial and open-source software.  The intern will have the chance to contribute to these projects by (1) adding complex physics to advanced simulations, (2) using these simulation tools to model the physics of sustainable manufacturing, and (3) creating reduced-order models that can be incorporated into design decisions.   Research will be tied to operating conditions defined by industrial partners. This is an emerging research topic with significant opportunities for future investigation and publication.

The intern can expect to build skills in the following areas:

1) High-performance computing.

2) Numerical analysis including computational fluid dynamics.

3) Energy systems modeling.

Interns in HPACF are encouraged to present at both local and national conferences, and to submit their research to peer-reviewed journals.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

Training a neural network involves selecting numerous model parameters (e.g., hidden-layer width and depth, weight and bias initialization, activation function choice, and network connectivity) and training-configuration parameters (e.g., optimizer choice, learning rate, regularization, tensor normalizations, and mini-batch size).  These choices can greatly impact the convergence behavior of the training process.  We have developed the Inference-Engine deep learning library to support the incorporation of neural-network-based model surrogates into high-performance computing applications.  Inference-Engine offers concurrent inference, the state-of-the art Adam optimizer, and also provides a platform for studying language-based parallel and GPU programming in Fortran 2023.  In our first target application, the Intermediate Complexity Atmospheric Research (ICAR) model, the cloud microphysics model that predicts precipitation occupies approximately 90% of the runtime.  We have recently demonstrated convergence of the training process for a neural-network that is now a candidate surrogate for ICAR's most widely used cloud microphysics model.  This project will involve studying techniques for tuning the aforementioned model parameters and training parameters to improve the convergence rate of neural-network training.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA

Project Description:

NREL’s Institutional Knowledge Graph project aims to organize institutional knowledge for collective scientific advancement. Until now, scientific knowledge has been decentralized across disparate databases, webpages, publications, and the minds of researchers. But with a graph containing the complex network of relationships between the projects, people, data, and tools that make up applied research at NREL, we can leverage what is known collectively to do better science.

Organizing NREL’s knowledge is just the first step. As part of the MSGI, the applicant will contribute to the development of a knowledge recommendation system, ensuring that researchers across NREL have access to data, tools, methods, and people who could be useful to their efforts. The successful applicant will learn to understand and implement graph completion and graph question-answering algorithms using state-of-the-art machine learning on heterogeneous directed graphs (for example, graph neural networks and graph attention networks). The applicant may also have the opportunity to contribute to a journal publication on the scientific knowledge recommendation system.

Disciplines: Applied Mathematics, Combinatorics, Computational Mathematics, and Probability and Statistics

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

The main objective of this project is to develop algorithms for data reduction of solutions to high-dimensional dynamical systems.  Specifically, under the direction of the mentor, the student will help design and implement hybrid methods for evolving low-rank approximations to solutions of partial differential equations (PDEs) --including kinetic models. The student will learn modern techniques for evolving low-rank data in a dynamical system; determine physical regimes where solutions to kinetic problems are low-rank through numerical simulations; become more familiar with PDE solvers and numerical discretizations for kinetic problems.  The student will give a presentation of the project at the end of the appointment.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

Continual learning in machine learning refers to the ability of models to learn continuously from new data, adapting to new tasks while retaining previously learned information, that are key for applications in edge computing and the continual adaptation of language models without full retraining. However, such models seem to be very sensitive to the order in which the tasks arrive. The goal of this project to develop robust optimization techniques to address the combinatorial challenges of task ordering through "Worst-Case Scenario Analysis" for optimal performance in challenging conditions, "Adversarial Training" to build resilience against unexpected task sequences, and "Uncertainty Modeling" to inform decision-making in dynamic environments.

Disciplines: Applied Mathematics, Operations Research, and Probability and Statistics

Hosting Site:

Argonne National Laboratory (ANL)

Internship location: Lemont, IL or virtual

Project Description:

The goal of this project is to investigate the behavior of numerical algorithms under the assumption of statistical variations and uncertainty in the computational state. With the emergence of new microelectronic materials and new system architectures, it is expected that the state of a computer system is no longer deterministic.  Rather the state may vary due to fluctuations in material properties or due to a lack of precision in the underlying algorithm.  To address these challenges, resilient and probabilistic computational models will be required.

In this project, the student will learn about probability, fundamental kernels in scientific computing, and basic numerical analysis.  The student will collaborate with research staff in mathematics and scientific computing and have the opportunity to present their research at the end of the summer.

Disciplines: Applied Mathematics, Combinatorics, and Computational Mathematics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

Topologically, electrical power grids can be treated as large-scale complex networks or graphs. Therefore, graph theory can be applied to analyze power grids to deal with the problems such as network decomposition, clustering, feeder routing, and dynamic microgrid formulation, etc. The physical motivations come from the enhancement of grid reliability, stability, and structural resilience with effective means.

This project will introduce mathematical PhD students with power grid network knowledges, survey and identify one of the most promising problems using directed graph theory, and perform a successful case study, e.g. decomposing a test grid with hundreds of nodes into multiple clusters based on certain criteria or features, using a programming language.

Disciplines: Algebra or Number Theory, Analysis, Computational Mathematics, Probability and Statistics, and Topology

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN

Project Description:

Forensic science's role in the legal system is pivotal. Yet, NAS (2009) and PCAST (2016), found that forensic methods in some pattern comparison disciplines -- such as fingerprints, firearms, footwear impressions, etc. -- lacked scientific validation. As a response to such criticisms, the forensic science community has been conducting studies, using ground-truth known scenarios, in which the forensic science practitioners and methods are tested in order to understand how well the systems perform. Such studies are generally called Black-box studies. Analyses of data from black-box studies aimed at assessing accuracy and reproducibility in pattern comparison forensic disciplines face a challenge when participants who sign up for the study either drop out or fail to respond to some or all of the test items. Some Bayesian hierarchical ""not missing at random"" models, developed outside of the forensic science context, have recently received attention and been applied to analyze data from two recently conducted black box studies. Results of these analyses suggest that the performance of forensic methods in pattern disciplines may be much worse than previously thought.

The goal of this project is to address the strengths and weaknesses of these hierarchical Bayesian models with primary focus on parameter identifiability and the effect of statistical modeling assumptions on the results. This will allow us to evaluate the trustworthiness of the results obtained from applying such models. This will have significant impact on how the forensic science community will assess the reliability of various forensic pattern discipline methods.

Disciplines: Analysis, Applied Mathematics, Biometrics and Biostatistics, Computational Mathematics, Geometry, Mathematics (General), Operations Research, Probability and Statistics, and Topology

Hosting Site:

National Institute of Standards and Technology (NIST)

Internship location: Gaithersburg, MD or virtual

Project Description:

Foundation models (FMs) open new opportunities for generalization of neural networks across different data modalities. FMs are large-scale deep learning neural network models (e.g., vision transformers) that are trained on vast amount of data and are expected to be tuned to a wide range of downstream tasks with relatively little additional task-specific training. Most FMs are currently restricted to two or three modalities such as text, RGB-color images and video. Transfer of FMs to processing other data modalities is an exciting direction for development of multi-modal FMs. This project will focus on adopting vision transformer models and transferring them to the interpretation of synthetic aperture radar (SAR) imagery. This is a fundamentally harder problem than optical imagery interpretation. SAR data are particularly valuable as SAR sensors can penetrate clouds and work in any illumination setting (day or night). As such, SAR image data could be particularly valuable in a disaster-response scenario where cloud cover and adverse conditions often limit the value of traditional optical imagery. The project will target development of transformer model to extract objects from SAR imagery. In particular, recently introduced SpaceNet-6 (Multi-Sensor All-Weather Mapping) challenge will provide a baseline and an open overhead imagery dataset acquired in two modalities over the same area, both quad-polarized X-band SAR and electro-optical imagery. The challenge is semantic segmentation, specifically extraction of building footprints. While model training can be done using both modalities, testing will have to be done using only SAR data. This is done to follow real-world scenario, when concurrent collection of elector-optical and SAR imagery is often not possible.

The participant will advance their knowledge in computer vision, machine learning, remote sensing and statistics. They will take an active role in developing transformer model(s) and approaches to transfer them to SAR data interpretation. The participant is expected to implement modelling in Python, PyTorch, and other machine learning and statistical related packages. They will also evaluate transformer model(s) on real-world aerial and satellite image data, will learn and use the appropriate evaluation metrics, and identify remaining challenges and how they can potentially be addressed. By learning alongside a multi-disciplinary team with expertise in statistics, machine learning, and remote sensing, they will gain expertise in technical communication and better appreciate team-based research efforts.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

U.S. Citizenship is a requirement for this internship

Project Description:

Inexact computing is any kind of computing where one does not get the exact numerical result. This can include approximate and probabilistic computation. This will be applicable to a wide range of post-Moore’s era architectures, because of reliability issues, potential power savings, increased resilience to faults and architectural changes. Some combination of general processors, general inexact processors and specialized inexact processors will have to be developed, as well as efficient ways to use them.

LANL has an ongoing exploration of inexact computing techniques, with projects in a range of areas of inexact computing. We are exploring reduced precision, machine learning approaches, advanced error detection and correction methods and other techniques, and applying these to problems in computational mathematics, basic mathematics and computer science. The specific project we address with an NSF-MSGI intern will depend on intern interests and background. Our current projects include:

1. An exploration of techniques from arithmetic combinatorics for integer problems, with application to novel devices.
2. Applications of machine learning to Boltzmann machines using an Ising model, and in particular, investigations of the fault model and detection and correction methods (perhaps using machine learning techniques) that may mitigate such faults.
3. Approximate matrix factorization for use in novel hardware.
4. Machine learning as applied to non-convex quadratic optimization problems.

>We encourage publication of results. LANL has a wide range of compute systems, and students will have access to cutting-edge devices of interest. If on-site activity is possible at the time of the internship, the intern will sit in the Ultrascale Systems Research Center, which supports a wide range of research in computer science.

We are happy to discuss the project in more detail upon request. For further information, please contact: Dr. Laura Monroe (lmonroe@lanl.gov).

Disciplines: Algebra or Number Theory, Applied Mathematics, and Mathematics (General)

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

U.S. Citizenship is a requirement for this internship

Project Description:

Disciplines: Algebra or Number Theory, Combinatorics, and Geometry

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM or virtual

U.S. Citizenship is a requirement for this internship

Project Description:

Understanding local change over time is a multifaceted effort that often requires an integrated approach. This process involves identifying and understanding shifts, transformation and developments within a specific geographic area at the neighborhood, town, or region scale.

As an intern, you will be part of a multidisciplinary team which focuses on understanding the interrelationship of earths physical features and human influence into a particular location by leveraging geospatial capabilities, image analysis, and advanced object-oriented change. The goal of this team is to extract information and knowledge about tangible features often found in urban/suburban environments such as roads, buildings, water, and high vegetation.

The primary responsibilities include preliminary data analysis and leveraging applied math frameworks to train and validate deep learning models in order to enhance model generalization. The intern will conduct training of multiple deep learning models, assess each model’s performance on predicting desired features, and carry out preliminary accuracy assessments using various statistical approaches. Additionally, the intern will identify areas for model improvement and finetuning. Lastly, the intern is expected to create clear and concise documentation of findings and model performance to the team.

Strong programming skills, preferably in Python, and familiarity with deep learning frameworks such as TensorFlow or PyTorch.

Knowledge of image processing techniques and computer vision principles.

Excellent problem-solving and analytical skills.

Effective communication and collaboration abilities.

Disciplines: Analysis, Applied Mathematics, and Computational Mathematics

Hosting Site:

U.S. Army Corps of Engineers, Geospatial Research Laboratory (GRL)

U.S. Citizenship is a requirement for this internship

Project Description:

This project connects machine-learning with machine-learning practice. Statistical bounds for nonlinear function approximation have not provided tools to accurately predict empirical performance of neural network models. As a result, practitioners rely on computational evaluation of models, which does not guarantee robust prediction accuracy. A recent focus has been on representing large parametric models with kernel formulations which open them up to analysis by well-established mathematical tools. At LANL, we have developed an exact Kernel based formulation for arbitrary Artificial Neural Networks (ANNS) which has a wide variety of applications both to produce theoretical guarantees, and to provide robust mathematical foundation to empirical methods in Out-of-Distribution (OOD) Detection, Adversarial Robustness, Signal Manifold Approximation, and more. In particular, these formulations allow decompositions of predictions into combinations of gradients attributed to each training input. This decomposition is implicitly the foundation of much of modern interpretability, OOD, robustness, and other techniques, and yet the connection between these techniques and these decompositions is only now being realized. Building upon our previous work, this project will further refine the mathematical theory for these representations and build approximation algorithms that can make near-exact Kernel surrogate models that can enhance the performance and reliability of large, real-world neural network models.

Disciplines: Applied Mathematics, Computational Mathematics, Foundations, Geometry, and Topology

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

Project Description:

We are developing efficient methods to solve inverse problems in fusion energy, enabling complex simulation codes to be routinely applied to experimental data with Uncertainty Quantification. The most complete physical models are computationally expensive and have a high-dimensional input space, but may be accelerated using initial conditions derived from lower fidelity models. We have successfully used this approach in specific cases and are now generalizing and automating these tools.

A student joining our multi-disciplinary team will learn both statistical techniques and applied mathematics relevant to plasma physics simulations of tokamak fusion devices. First they will learn Bayesian Optimization methods for high-dimensional problems, building on existing examples and tools. He or she will then evaluate the impact of model fidelity on computational efficiency, and learn methods for mapping between model fidelities to accelerate high-dimensional inverse problems. Depending on results, we anticipate that by the end of the summer we will write a journal publication together.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Livermore National Laboratory (LLNL)

Internship location: Livermore, CA or virtual

Project Description:

The objective of this project is to investigate how the design of cost functions affects the trainability of variational quantum algorithms. The near-term quantum processors are noisy, and errors vastly limit their ability to execute lengthy algorithms. Variational quantum algorithms offer a promising potential toward practical quantum advantage on several specific applications utilizing noisy quantum processors without error correction. The error tolerance comes from offloading part of the work to classical optimization of cost functions measured by quantum processors. The cost functions are generally non-convex, and their gradients may decrease exponentially with increasing system size. Therefore, the trainability of the cost functions is one of the most significant challenges for achieving quantum advantage using the variational approach. In this project, we will explore how different choices of cost function design, e.g., the operators to measure, affect trainability and how optimization strategies can be adjusted accordingly.

Physics knowledge and quantum computing are not required for this project. Students should be comfortable with optimization problems and have some experience (or a willingness to learn) programming in Python or Julia.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Fermi National Accelerator Laboratory (FNAL)

Internship location: Batavia, IL or virtual

Project Description:

This project aims to develop efficient protocols for synthesizing qudit gates across diverse hardware implementations. Qudits are multi-level quantum systems that can serve as fundamental units for encoding information in quantum computing. Compared to qubits, which have two levels, qudits can offer advantages in specific applications by executing certain computational operations more efficiently. Synthesizing efficient qudit gates requires optimizing native gate or pulse sequences. This optimization task is generally challenging due to the non-convex and barren plateau cost function landscapes, especially when dealing with a large number of qudit levels. An efficient protocol for synthesizing qudit gates is crucial for the success of quantum computation utilizing qudits. This project will investigate the performance and optimization complexity of qudit gate synthesis protocols. The outcomes will provide valuable insights into the design of qudit gates and their control protocols.

Physics knowledge and quantum computing are not required for this project. Students should be comfortable with linear algebra and optimization problems and have some programming experience (or a willingness to learn).

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Fermi National Accelerator Laboratory (FNAL)

Internship location: Batavia, IL or virtual

Project Description:

Background and Learning Experience:

This project is a unique and exhilarating journey into the intersection of applied mathematics, machine learning, and cybersecurity within the realm of Cyber-Physical Systems (CPS). Participants will have the opportunity to be at the vanguard of technological innovation, focusing on enhancing the robustness of ML algorithms against adversarial attacks in critical CPS domains. It offers a rich blend of theoretical exploration and hands-on practical experience, allowing participants to delve deep into the intricacies of securing ML models in an increasingly interconnected and technology-driven world. By engaging in this project, participants will not only gain invaluable knowledge and skills in a cutting-edge field but also contribute to shaping the future of technology, making them invaluable assets in sectors ranging from autonomous transportation to smart infrastructure. This project isn’t just an educational endeavor; it's an adventure at the forefront of technological advancement, offering a unique chance to tackle some of the most pressing cybersecurity challenges of our time.

Learning Objectives:

Participants will:

1. Explore Mathematical Concepts: Understand and apply mathematical theories to cybersecurity challenges in CPS, enhancing their knowledge in applied mathematics and probability theory.
2. Develop and Test Algorithms: Gain experience in creating and refining algorithms designed to protect ML models from adversarial threats. This will include learning about algorithm design, optimization, and testing in simulated environments.
3. Collaborative Learning: Engage in interdisciplinary collaboration, gaining insights into the integration of cybersecurity, machine learning, and specific CPS applications.
4. Research and Analysis: Conduct research and analyze data to understand the impact of adversarial attacks in CPS, fostering skills in critical thinking and problem-solving.
5. Communication Skills: Develop and enhance communication skills through the creation of a research paper and educational materials, learning to convey complex technical information effectively.

Activities and Participation

Participants in this project will have the opportunity to:

1. Engage in the development of a comprehensive mathematical framework to understand and counter adversarial threats in ML models.
2. Participate in the creation and iterative refinement of robust ML algorithms through hands-on application and testing.
3. Collaborate with experts in related fields, gaining interdisciplinary knowledge and experience.
4. Contribute to the simulation and testing phase, applying theoretical knowledge in practical scenarios.
5. Assist in the preparation of a scholarly article, presentation, and practical implementation guides, enhancing their writing and communication skills.

Impact: This project aims not only to contribute significantly to the field of cybersecurity in CPS but also to provide participants with a comprehensive learning experience. By engaging in this project, participants will enhance their understanding and skills in applied mathematics, machine learning, cybersecurity, and the practical application of these disciplines in real-world scenarios.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

Background and Learning Experience

This exploratory project offers a unique foray into the application of Generative Artificial Intelligence (AI) within the realm of cybersecurity, particularly focusing on the protection of critical infrastructure. It stands at the exciting intersection of AI innovation and cybersecurity, a field that is rapidly gaining significance in our increasingly digital world. Participants will delve into the world of generative models, exploring how these advanced AI tools can be leveraged to predict, detect, and counter cyber threats against essential infrastructure systems. This project not only offers a chance to engage with cutting-edge AI technology but also to apply these innovations in practical, real-world scenarios that have a direct impact on national and global security.

Learning Objectives

Participants will:

1. Explore Generative AI Models: Understand and engage with the latest developments in generative AI, including deep learning techniques like Generative AI's.
2. Innovative Application in Cybersecurity: Learn how to apply these AI models to simulate potential cyber threats and develop robust defense strategies for critical infrastructure systems.
3. Cross-Disciplinary Exposure: Gain exposure to a blend of disciplines, including artificial intelligence, cybersecurity, and infrastructure technology, fostering a comprehensive understanding of this interdisciplinary field.
4. Practical Problem-Solving: Develop skills in applying theoretical AI concepts to practical cybersecurity challenges, enhancing problem-solving and analytical abilities.
5. Communication and Collaboration: Enhance communication skills through collaborative reserearch and the development of materials that explain complex AI concepts in accessible ways.

Activities and Participation:

Participants in this project will have the opportunity to:

1. Engage in the development and training of generative AI models, understanding their potential applications in cybersecurity.
2. Participate in innovative research to apply these models for the prediction and simulation of cyber threats against critical infrastructure.
3. Collaborate with a team of experts in AI and cybersecurity, gaining valuable insights and contributing to interdisciplinary solutions.
4. Contribute to the creation of strategies and protocols for implementing AI-driven cybersecurity measures in real-world scenarios.
5. Assist in the documentation and dissemination of research findings and methodologies, enhancing their technical writing and presentation skills.

Impact

This project is designed to not only advance the field of AI in cybersecurity but also to provide participants with a rich and comprehensive learning experience. Participants will be at the forefront of exploring how generative AI can revolutionize the way we protect our most critical digital and physical assets, preparing them for future challenges and opportunities in this vital field.

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

Background and Learning Experience

This innovative project introduces the concept of 'Prompt Engineering' as a tool for enhancing cybersecurity in Cyber-Physical Systems (CPS). Prompt engineering, primarily used in the realm of advanced AI and machine learning, involves crafting precise and effective input prompts to elicit desired responses or behaviors from AI systems. In the context of CPS cybersecurity, this technique can be pivotal in identifying, mitigating, and defending against sophisticated cyberattacks. Participants in this project will explore the intersection of AI, cybersecurity, and CPS, gaining hands-on experience in using prompt engineering to strengthen the security of critical infrastructure against cyber threats.

Learning Objectives

Participants will:

1. Understand Prompt Engineering: Acquire knowledge about prompt engineering and its applications in AI and cybersecurity.
2. Apply Generative AI Techniques in Cybersecurity: Learn how to utilize Gen AI and machine learning techniques, particularly prompt engineering, to anticipate and counter cyber threats in CPS cybersecurity.
3. Innovative Problem-Solving: Develop skills in crafting effective prompts to simulate, detect, and respond to potential cyberattacks, enhancing critical thinking and creative problem-solving abilities.
4. Effective Communication: Improve communication skills through collaborative projects and presentations, learning to convey complex technical concepts in a clear and accessible manner.

Activities and Participation

Participants will have the opportunity to:

Participants will engage in:

1. Designing and refining prompt engineering techniques for cybersecurity applications within CPS.
2. Implementing and testing these techniques in simulated environments to evaluate their effectiveness against various cyber threat scenarios.
3. Analyzing the outcomes and refining strategies based on AI responses to engineered prompts.
4. Collaborating with a team of experts in AI, cybersecurity, and CPS to develop comprehensive, cutting-edge solutions.
5. Documenting and sharing research findings, contributing to the broader knowledge base in this specialized field.

Impact

This project aims to pioneer the use of prompt engineering in cybersecurity, offering participants a unique chance to contribute to an emerging field. The knowledge and skills gained will be crucial in advancing cybersecurity measures for CPS, preparing participants for future roles in safeguarding critical digital and physical infrastructures.

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

Background and Learning Experience:

This pioneering project investigates the phenomenon of hallucinations in Large Language Models (LLMs) and their potential exploitation by cyber adversaries to initiate attacks on Cyber-Physical Systems (CPS). Hallucinations in LLMs refer to instances where these models generate incorrect, often nonsensical or unrelated information, which can be a significant vulnerability in cybersecurity. Participants in this project will explore this less-charted territory, understanding how LLM hallucinations can be manipulated for cyberattacks and developing strategies for prevention and mitigation.

Learning Objectives

Participants will:

1. Understand LLM Hallucinations: Acquire a comprehensive understanding of the causes and nature of hallucinations in LLMs and their implications in cybersecurity.
2. Identify Vulnerabilities: Learn to identify how these hallucinations can be exploited by cyber adversaries to compromise CPS.
3. Develop Prevention Strategies: Gain skills in developing strategies and techniques to prevent the manipulation of LLM hallucinations in initiating cyberattacks.
4. Mitigate Risks: Understand and implement risk mitigation strategies for potential cyber threats arising from LLM vulnerabilities.
5. Engage in Cross-Disciplinary Research: Collaborate in a cross-disciplinary environment, integrating knowledge from AI, machine learning, and cybersecurity to address complex challenges in CPS.

Activities and Participation

Participants will:

1. Engage in research to understand the phenomenon of LLM hallucinations and their potential exploitation by cyber adversaries.
2. Participate in simulations and scenario analysis to identify vulnerabilities in CPS arising from LLM hallucinations.
3. Develop and test strategies for the prevention of such vulnerabilities and the mitigation of their impacts.
4. Collaborate with a team of experts in AI, machine learning, and cybersecurity, contributing to innovative and cross-disciplinary solutions.
5. Document and share findings, enhancing technical writing and knowledge dissemination skills.

Impact: This project aims to contribute significantly to the understanding of a novel aspect of AI security, particularly in the context of CPS. Participants will not only gain specialized knowledge and skills but also contribute to advancing the field of cybersecurity, addressing emerging threats in an increasingly AI-integrated world.

Disciplines: Applied Mathematics, Computational Mathematics, Foundations, and Probability and Statistics

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

Background and Learning Experience:  This project is centered on the generation and validation of synthetic data for cybersecurity applications in Cyber-Physical Systems (CPS). It aims to address a critical challenge in the field: ensuring that the decisions made by models trained on synthetic data are applicable and effective in real-world scenarios. Participants will explore innovative methods for creating realistic synthetic datasets that can safely and effectively simulate cyber-physical environments, as well as develop validation frameworks to assess the feasibility and reliability of these datasets in real-world applications.

Learning Objectives

Participants will:

1. Synthetic Data Generation: Learn advanced techniques for generating high-fidelity synthetic data that accurately simulates the complexities of real-world CPS environments.
2. Model Training and Testing: Gain experience in training cybersecurity models using synthetic data and testing their effectiveness in simulated scenarios.
3. Validation Techniques: Develop skills in creating and implementing validation frameworks to assess the applicability of decisions made by models trained on synthetic data in real-world CPS scenarios.
4. Feasibility Analysis: Understand how to analyze the feasibility and reliability of model decisions when applied to actual cyber-physical systems, considering factors like data fidelity, model robustness, and real-world variability.
5. Cross-Disciplinary Collaboration: Engage in interdisciplinary research, combining aspects of AI, cybersecurity, data science, and CPS engineering.

Activities and Participation

Participants will:

1. Engage in the development of algorithms and methods for generating synthetic data that closely mimics real-world CPS environments.
2. Train cybersecurity models using this synthetic data and test their performance in simulated scenarios.
3. Create and implement validation frameworks to assess the real-world applicability of these models.
4. Analyze the decision-making feasibility of these models when applied to actual CPS, identifying gaps and areas for improvement.
5. Collaborate with experts across various fields, contributing to a comprehensive approach to CPS cybersecurity.

Impact: This project aims to significantly advance the field of CPS cybersecurity by addressing one of its key challenges: the development of reliable and applicable synthetic data for model training and validation. Participants will contribute to enhancing the safety and security of CPS by ensuring that cybersecurity measures are robust and effective in real-world conditions.

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

National Renewable Energy Laboratory (NREL)

Internship location: Golden, CO or virtual

Project Description:

AI models are gaining wide interest within the medical community due to their increasing ability to rapidly assimilate text in the form of electronic health records (EHR) and predictive skill. The predictive performance of any AI model is dependent on the choice of model architecture and hyperparameters, such as the specific choices of activation function in each layer, learning rates, batch sizes, and choice to fine-tune the word embeddings (or not) during model training. Indeed some sets of hyperparameters may lead to overfitting or underfitting a model. As a direct consequence, these models will have poor predictive performance and/or lack of generalization outside of the training dataset. Consequently, identifying an appropriate set of hyperparameters is crucial to ensure the model’s accuracy and generalizability.

Identifying a  hyperparameter set that would make our NLP models both more accurate and robust requires a prohibitively expensive search over a high dimensional space, as large models must be trained and validated at each hyperparameter combination. High performance computing systems may be leveraged to alleviate some the computational burden of this search by exploring multiple hyperparameter sets simultaneously. Due to the coupled nature of some hyperparmeters however, this problem is not trivially parallel.

As part of this project, we will develop and implement scalable hyperparameter optimization (HPO) algorithms from the clinical NLP library, FrESCO, deployed as part of the EHRLICH project, which may include: a multi-task convolutional neural network, multi-task hierarchical self-attention network, and a clinical Transformer architecture. We will use the RayTune library for hyperparameter tuning and will integrate these RayTune functionalities as modules into our FrESCO clinical NLP library. The performance of our HPO procedure will be assessed in terms of: (1) scalability attained by distributing the hyperparameter search over multiple compute nodes at the Oak Ridge Leadership Computing Facility (OLCF) and (2) validation accuracy of the trained clinical NLP model on MIMIC-IV data.

The expected outcome is a scalable HPO framework integrated within the existing FrESCO library which attains linear scaling over multiple compute nodes on the OLCF Frontier supercomputer, with an improved accuracy or decreased time to solution as compared with existing clinical NLP models trained on the same EHR dataset.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak RIdge, TN

U.S. Citizenship is a requirement for this internship

Project Description:

This project is to work with the original Fortran 77 code developed by Norman Lebovitz and Alexander Lifschitz to perform a high resolution numerical ellipsoidal harmonic instability analysis of the Riemann S-type ellipsoids. The Riemann S-type ellipsoids are uniform density (incompressible) inviscid self-gravitating equilibrium triaxial rotating fluid ellipsoids. These ellipsoids which have been studied for many centuries were popularized by Nobel Laureate Subrahmanyan Chandrasekhar in his 1969 book "Ellipsoidal Figures of Equilibrium". It was noticed by Chandrasekhar and Lebovitz (Chandrasekhar's student) that these equilibrium fluid ellipsoids become unstable to a dynamical shape changing instability as the angular momentum of the ellipsoids (eccentricity) is increased. Using a numerical code originally developed by Lebovitz and Lifschitz, we would like to further explore the beautiful properties of these nonlinear instabilities by exploiting the original Lebovitz-Lifschitz code. To find more detail about the instability analysis, code and algorithms which are due to Norman Lebovitz (d.2022) and Alexander Lifschitz (now Alexander Lipton), see the following papers: (1) Lebovitz & Lifschitz, New Global Instabilities of the Riemann ellipsoids, The Astrophysical Journal (1996) 458, 699-713; (2) Lebovitz & Lifschitz, Short-wavelength instabilities of Riemann ellipsoids, Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical Sciences and Engineering, 354 (1996), no. 1709, 927–950.

Disciplines: Applied Mathematics

Hosting Site:

National Institute of Standards and Technology (NIST)

Internship location: Gaithersburg, MD or virtual

Project Description:

A long-standing problem in the electricity sector is the lack of transparent reflection of outage and insufficiency risks in energy pricing. This leads to mostly flat yet spiky price signals, with sudden price spikes arising during extreme scarcity, as observed during the 2021 winter storms in Texas. To mitigate these price spikes, policymakers introduce ancillary products, such as capacity markets and operating reserve curves, aimed at providing more stable price signals to infrastructure investors. These ancillary products, however, are often criticized for their discretionary nature. Their quantities and prices are computed based on idiosyncratic formulas that do not adapt to new trends or technologies. For example, conventional capacity markets disregarded renewables energy resources as suppliers until recently.

Our approach seeks to address this issue by utilizing risk measures, such as Conditional Value at Risk (CVaR), to assess the impact of potential outages or sufficiency events during normal operating conditions. This approach offers the advantages of being technology-agnostic, requiring only the setting of two parameters based on historical engineering standards, e.g., 1 day/10 years security criterion. It directly informs energy pricing through duality, without the need for additional products or services. The main challenge of the proposed approach lies in computational complexity: optimizing power scheduling problems with risk measures can be complicated, especially when handling industrial-scale power scheduling problems.

To address this challenge, we will expand on recent research on low-complexity algorithmic approaches for CVaR-constrained optimization problems. We will adapt these algorithms to a CVaR-constrained version of the security-constrained DC (linearized) optimal power flow problem (CVaR-DCOPF) where we restrict the total load loss across the worst k contingencies to be less than certain pre-specified quantity. We will implement a parallel numerical solver for CVaR-DCOPF, building on prior tools developed by the team at LLNL, and test the performance of the approach on industrial-scale instances. On the theoretical side, we will investigate techniques to replace the projection operator in current algorithms by functional constraints, which allow us to capture nonlinear problems in future research. Finally, we will numerically study the effect of parametrizing CVaR constraints in energy prices and assess the impact of such prices on investment over the long term, to inform the viability of the approach to ensure security of supply of electricity to end costumers.

As an intern on this project, the applicant will have the opportunity to learn the cutting-edge research at the intersection of energy economics and computational optimization. The learning goals involve acquiring hand-on experience in:

1. Understanding the fundamentals of CVaR and their application in electricity pricing and infrastructure investment decisions.
2. Familiarizing themselves with algorithmic approaches for CVaR-constrained optimization problems and their implementation.
3. Exploring the complexities of power scheduling problems, focusing on CVaR-DCOPF problems and their solutions.
4. Participating in the development and refinement of a parallel numerical solver for CVaR-DCOPF, leveraging existing tools and methodologies pioneered by the team at LLNL.
5. Collaborating with experienced researchers to conduct analyses on the impact of CVaR constraints on energy prices and long-term investment decisions in the electricity sector.

Throughout the internship, the selected candidate will engage in diverse activities, including data analysis, algorithm development, software implementation, and collaborative research discussions. This experience will provide valuable insights into real-world energy challenges and equip the intern with practical skills in computational optimization, energy economics, and algorithm development.

Disciplines: Analysis, Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Livermore National Laboratory (LLNL)

Internship location: Livermore, CA or virtual

Project Description:

Understanding the folding of biopolymers (including nucleic acids and proteins) is vital to ongoing research in biochemistry and bioengineering. Recent models built on deep learning can often accurately predict equilibrium structures. Nevertheless, full inference and summation in the Boltzmann ensembles of these systems is still an outstanding and vital research goal. If the ensemble of folding topologies is suitably restricted, algorithms based on dynamic programming can achieve exact summation in relatively low cost. We are seeking to extend the applicability of these algorithms to general folding topologies using ideas from random matrices and matrix product states. A student in this internship will have the opportunity to learn sophisticated theories related to polymer physics, random walks, tensor networks, and random matrices with an eye towards efficient algorithm design and computational implementation.

Disciplines: Applied Mathematics, Computational Mathematics, and Mathematical Biology

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA or virtual

Project Description:

Current and near-term quantum computers, often referred to as noisy intermediate-scale quantum (NISQ) computers, are characterized by low qubit counts, short qubit decoherence times, and high gate error rates. Despite these limitations, there is significant promise in harnessing NISQ computers through the use of hybrid quantum-classical algorithms. These algorithms leverage both quantum and classical hardware to perform specific computational tasks. The goal of this project is to leverage the capabilities of NISQ quantum computers to estimate eigenenergies in various scientific domains, including physics, chemistry, and materials science. Eigenenergies are critical in understanding the behavior of quantum systems, such as molecules and materials, and they have wide-ranging applications in predicting properties, reactions, and behaviors. We aim to develop approaches that are resilient to noise and can accurately estimate eigenenergies from quantum dynamics.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA

Project Description:

Most particle transport codes fall into two major categories – Monte Carlo (MC) codes, that rely on stochastic simulations, and deterministic codes, that rely on numerical approximation and discretization methods. MC codes are memory-bound, being limited by the enormous memory footprints arising from the need to store and read from the memory very large data sets describing simulation particles. In the deterministic transport one of the major computational bottlenecks is the ‘curse of dimensionality’ of the underlying quadrature integration routines, such as the Discrete Ordinates, or Sn, integration method. In both instances  thermal radiation transport (TRT)  codes can be dramatically improved when the underlying methods are coupled to the latest statistical and machine learning (ML) algorithms. The goal of this project is to explore advanced statistical learning methods, such as the Stochastic Expectation Maximization, Gibbs Sampling, and Markov Chain Monte Carlo, in order to address two distinct problems arising in TRT simulations: memory usage reduction in Monte Carlo TRT codes, and reduction of computational complexity of the deterministic TRT codes.

Disciplines: Applied Mathematics, Computational Mathematics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

U.S. Citizenship is a requirement for this internship

Project Description:

USACE reservoirs are experiencing more frequent and more severe algal blooms.   Harmful algal blooms (HABs) , those that include toxin producing cyanobacteria and phytoplankton species, are also increasing.   These disruptive events are believed to be caused by changes in flow and thermal conditions as well as nutrient water quality that are induced by the combined effects of climate and land use change.  In short, the loading rates and internal storage of critical factors in algae growth such as nutrients, carbon, and sediments are changing concurrently with precipitation, wind, and air temperature.  Thus, the relationships between algal growth and water quality parameters that have traditionally been applied to reservoirs can be outdated in terms of parameter value ranges and fit to individual reservoirs. 

Planned activities

The participant will utilize historical multi-parameter water quality observations to develop statistical tools for deriving site specific driver relationships to be used in numerical water quality models.  This will involve compiling and preparing data sets, developing sorting algorithms, applying multiple types of regression analysis (linear, quantile, etc.), analyzing for common patterns, and documenting the process and results.

Necessary skills

The intern should have experience with data handling; algorithm development, coding, and scripting; multivariate statistical analysis; and stochastic modeling.  An interest in complex natural systems is preferred but not required.  Use of R is required. Experience with Python, Excel, Visual Basic, and MATLAB may also be helpful.  This position requires formal scientific writing.  The participant will interact regularly within a water quality engineering team and have opportunities to learn about ongoing research in other emerging water quality topics such as wildfires, PFAS, and microplastics.

Disciplines: Analysis, Applied Mathematics, Mathematics (General), and Probability and Statistics

Hosting Site:

U.S. Army Corps of Engineers, Engineer Research and Development Center (ERDC)

Internship location: Vicksburg, MS or virtual

Project Description:

Partial differential equations (PDEs) that involve inequality constraints on the solution, the so-called obstacle problems, arise in various applications, including contact mechanics, phase-field modeling in material science and also mathematical finance. To accurately simulate these models there is a need to develop efficient and accurate discretization methods. The aim of this project is to design approximation algorithms for the obstacle-type problems using a neural network discretization approach. Under the guidance of the mentor, the student will help design and implement the neural network architecture for the approximation of the elliptic and time-dependent obstacle problems. During the internship, the student will learn about the theory of obstacle-type problems, discretization of such models and its use in scientific applications. The student will be able to interact and engage with other staff members as well as to give a presentation about the summary of the internship project at the end of the appointment.

Disciplines: Analysis, Applied Mathematics, and Computational Mathematics

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak Ridge, TN or virtual

U.S. Citizenship is a requirement for this internship

Project Description:

This project is dedicated to optimizing the applicability of fiber optic sensors (FOS) in industrial monitoring, with a specific focus on addressing challenges related to modeling the behavior of these sensors in the presence of physical defects observed in pipelines, such as corrosion, cracks, and welding issues. Despite the inherent advantages of fiber optic sensors in precisely measuring various parameters, the need for an effective simulation platform for modeling their behavior under different scenarios has become evident.

To tackle this challenge, the project aims to develop a sophisticated simulation platform employing scientific machine learning methodologies. This platform will specifically focus on modeling guided wave propagation on pipelines to simulate various physical defects. The simulation platform will generate synthetic data, which, when combined with experimental data, will be used to train machine learning models. These models will contribute to the development of an overall health index for the pipeline. Moreover, the simulation platform will be extended to create a digital twin of the gas pipeline, replicating real-life defects and gas transport conditions.

The primary objectives include the creation of an efficient real-time simulation platform for modeling physical defects in pipelines under various scenarios. The project will explore recent advancements in scientific machine learning methodologies, such as physics-informed neural networks and Fourier neural operators. This initiative represents the initial step towards establishing a comprehensive digital twin of a pipeline. Through these efforts, the project aims to unleash the full potential of fiber optic sensors, transforming them into highly effective monitoring tools across diverse industries, including aerospace, defense, security, civil engineering, energy, and healthcare.

Project Objectives:

Week 1 & 2:

Intern orientation with relevant people from the team. The candidate will be provided with materials and ongoing research related to the project for their learning and understanding of the project. Intern will also do some study on existing platforms for scientific machine learning.

Week 3&4:

The intern will finalize the platform to research upon such as nvidia modulus or deepxde..etc. They will focus on some tutorial problems similar to guided wave propagation.

Week 5&6:

The intern will implement the learned methodologies towards replicating guided wave propagation problem.

Week 7&8:

The intern will continue to investigate and research the problem adding physical defects.

Week 9&10:

Iterate on the model with feedback from team members and debug for any potential issues.

Finalize the results, prepare the relevant documentation and reports. Present the findings to the overall team.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

National Energy Technology Laboratory (NETL)

Project Description:

Preconditioning and solving large-scale linear systems underlies nearly all large-scale science and engineering simulations. There are approaches that work well for many cases like diffusion and electromagnetics, but there are many difficult problems such as turbulent flow and magnetohydrodynamics which require care to solve effectively.

Multigrid methods are a class of optimal computational complexity solvers and preconditioners which solve a linear system by recursively coarsening the problem size and improving the solution estimate. Construction of a coarse problem (also referred to as a coarse grid) may use geometric data via geometric multigrid (GMG) or use algebraic data from the matrix operator via algebraic multigrid (AMG). The interpolation between grids is supplemented by simple solver iterations called smoothers.

Interested participants will learn how multigrid methods interact with partial differential equations and discretizations through many examples involving practical applications. Participants will develop mathematical theory and practical intuition for improving linear solvers for a variety of problems. The team will collaborate to develop effective and performant approaches for difficult applications.

This project will utilize the parallel C++ library MueLu in the Trilinos framework (https://github.com/trilinos/Trilinos) for multigrid methods on Linux supercomputers, including some of the largest supercomputers in the world. Additionally, a Sandia summer proceedings article will be written about the completed project.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Sandia National Laboratories (SNL)

Internship location: Albuquerque, NM

Project Description:

Finite element methods (FEMs) are some of the most ubiquitous approaches for discretizing and solving partial differential equations (PDEs). FEMs allow one to construct finite dimensional space approximating an infinite dimensional space and solve a corresponding linear system arising from a PDE.

FEMs are utilized for a wide variety of applications, from diffusion and electromagnetics to turbulence and magnetohydrodynamics. In such applications, there are often additional desired properties such as fluid incompressibility or solenoidal magnetic fields. However, many multiphysics applications are constrained in size due to memory limitations. To that end, techniques such as basis compression and dimension reduction are applied in order to reduce computational burden and increase simulation speed.

Interested participants will learn how FEMs are used to discretize partial differential equations and how they are used to solve multiphysics applications. Participants will then help develop numerical analysis for FEM basis compression and software to validate the theory. The team will collaborate to advance the state of the art in basis compression for FEMs.

This project will utilize the parallel C++ library MrHyDE (https://github.com/sandialabs/mrhyde) developed utilizing the Trilinos framework (https://github.com/trilinos/Trilinos) for finite element methods on Linux supercomputers, including some of the largest supercomputers in the world. Additionally, a Sandia summer proceedings article will be written about the completed project.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Sandia National Laboratories (SNL)

Internship location: Albuquerque, NM

Project Description:

Flooding is the natural hazard causing major economic and social impacts worldwide. Urban hydrology includes complex interaction of surface flows, subsurface infiltration and flows in sewer drainage networks. Urban flooding often occurs when rainfall overwhelms the sewer system, causing water to pool in streets, yards, and house basements.Accurate modeling of urban hydrology is of utmost importance to understand water movement in urbanized areas and perform mitigation strategies.

In this project, the development and simulation will be conducted using multi-lab project Advanced Terrestrial Simulator(ATS) to provide efficient and accurate process-based model for urban hydrology. The model will include overland flows (diffusive wave and/or shallow water models), subsurface flows (Richards equation) and pipe network flows (shallow water 1-D Saint-Venant equation). Alongside a mentor, the successful applicant will have an excellent opportunity to perform research on advanced spatial and time discretization methods that have to be applied to perform high-fidelity predictive simulation at scale.

He/She will have the chance to enhance their knowledge on a wide range of fields including integrated hydrology modeling, numerical methods, and computational physics as well as high performance computing.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Los Alamos National Laboratory (LANL)

Internship location: Los Alamos, NM

Project Description:

Maxwell's equations are an important modeling component in a variety of modern engineering applications, ranging from nanoscale optical devices to magnetic confinement fusion.  Despite the significant advances in the mathematical analysis and numerical methods for the solution of Maxwell's equations, little progress has been made in our understanding of optimal control and optimal design problems governed by Maxwell's equations.  This project seeks to develop rigorous, robust and efficient solution methods for such optimization problems, to help realize the full potential of modern electromagnetic devices and systems.

The project will focus on optimal control, in both space and time, of electromagnetic sources (currents).  First, a rigorous mathematical formulation will be developed and analyzed.  The analysis will combine elements of semigroup theory, to handle the temporal aspects of the problem, with the de Rham complex for the gradient, curl and divergence operators, for a proper spatial setting.  The analysis will inform the problem formulation in the sense of defining an appropriate control regularization strategy.  Second, the problem will be discretized in space and time, using finite element spaces and time steppers that yield robust and accurate optimal controls.  Third, the problem will be solved numerically using Sandia's open-source software MrHyDE (https://github.com/sandialabs/MrHyDE), which combines a variety of scientific computing tools developed in the Trilinos project (https://github.com/Trilinos).

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Sandia National Laboratories (SNL)

Internship location: Albuquerque, NM

Project Description:

Density Functional Theory (DFT) is a computational quantum mechanical modeling approach used in fields like physis, chemistry and material science to investigate the structure of many-body systems, focusing on atoms and molecules and their interactions. DFT is among the most popular and versatile methods available in condensed matter physics, computational physics and computational chemistry. This is emphasized by the fact that more than 25% of the HPC workloads on leading supercomputing centers is spent in performing DFT simulations. Therefore, understanding the DFT algorithms and providing scalable and highly efficient implementations for these algorithms becomes mandatory, especially now when GPU and customized hardware accelerators are used to provide better performance and execution time. Building frameworks that allow for DFT codes to be executed on a myriad of systems, offering automatic optimizations, can provide crucial benefits to scientists working on problems that are becoming larger and more complicated.

Goals:

Provide an in depth study and comparison between different algorithms for the eigensolver problem. Most DFT calculations have an eigenvalue solver as their main computation. Depending on the system some algorithms may perform better than others, as such we would like to study three of these algorithms (Conjugate Gradient, Davidson, RMM) and understand their overall performance and convergence rate.

Provide an automatic mechanism to parallelize and distribute the data and computation across the network. All eigensolvers require linear algebra and Fourier transforms (in the case of planewave Density Functional Theory). These components may exhibit different communication patterns, which will affect the overall performance on a distributed setting. We want to investigate that and build models to guide our implementation using information from algorithms and the hardware.

Each algorithm may have different convergence rates. The three mentioned algorithms are iterative, and may require multiple iterations to converge to a solution. As such we would also like to investigate the different convergence rates and maybe offer solutions by implementing better preconditioners. This will add another dimension to our model.

Approach and methodology

Our study will concentrate on three key aspects:

1. Algorithm Performance Assessment: We want to evaluate and compare the performance of each algorithm. This includes their efficiency , speed, and resource utilization. Most algorithms are implemented in C/C++ language and make use of MPI. The main goal is to run these codes on Perlmutter and devise a baseline. From there we can start modifying the communication and computation and evaluate the performance changes.
2. Convergence Analysis of Algorithms: We will focus on the convergence properties of the overall algorithm. This will involve studying how quickly and effectively these algorithms converge to a solution. We plan to talk to people working on actual problems from physics and material science. We will use some of the systems they are working on while running our experiments.
3. Development of Preconditioners: Here, we want to create and implement preconditioners to enhance the convergence rate of these algorithms. We will explore various preconditioning techniques and assess their impact on the speed and accuracy of convergence. Because the systems under analysis are going to be represented as matrices, each system will have different properties which translate to different shapes and structures in matrix notation. These structures can impact convergence and hence developing appropriate preconditioners for this will be critical.

Disciplines: Applied Mathematics, and Computational Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory (LBNL)

Internship location: Berkeley, CA

Project Description:

In this project we will develop and study stochastic models describing the growth and spread of pathogens in plant-pathogen interactions.  The models will form part of a digital twin framework for plant health monitoring.  In addition to working with the mentor in defining the models, project tasks will include implementation of numerical solution methods for systems of stochastic differential equations.

During the internship the student will gain experience with numerical methods which may include neural network approaches combined with traditional numerical methods for solving differential equations.  Python programming will be required.  Therefore, the student will be expected to learn Python programming during the course of the internship if he/she does not already meet this requirement.

Disciplines: Applied Mathematics, Computational Mathematics, and Mathematical Biology

Hosting Site:

Brookhaven National Laboratory (BNL)

Internship location: virtual

Project Description:

The discovery of matrix multiplication schemes has been a recent hot topic, and resulted in a few record breaking results for multiplying small matrices. In particular, the study of discrete objects known as 'flip graphs', which consists of sets of orbits of multiplication schemes and edges representing actions on those schemes, enables a way to perform searches for new ways to multiply two matrices. Thus far, these searches have been naively pursued and only on CPUs, which presents limitations very quickly in the large search space. The goal of this project will be to study how to adapt these flip graphs and develop algorithms that would effectively perform the search on GPUs, which would leverage the supercomputing power available to us at ORNL.

Learning Objectives:

1. Student will learn about new ways to discover matrix multiplication schemes for small sizes by reading the state-of-the art on the topic (papers published within the last two years).
2. Student will increase their algebraic and graph theory knowledge by studying the underlying mathematical structures behind flip graphs.
3. Student will learn about different computing architectures by assisting with the algorithmic design for speeding up search for new matrix multiplication schemes on GPUs.
4. Student will learn how to communicate scientific results by writing/speaking about the topic and reporting any results that they may be able to get during our ten weeks together.
5. Student will expand their mathematical and scientific network by participating in program activities, attending local seminars, and meeting with international collaborators studying the same topic.

Disciplines: Applied Mathematics, Computational Mathematics, and Foundations

Hosting Site:

Oak Ridge National Laboratory (ORNL)

Internship location: Oak RIdge, TN or virtual

Project Description:

Machine learning (ML) has shown great success in many domains, and increasing efforts are applying ML in the field of computer science and engineering, for instance, to design more efficient hardware and software. Due to its data driven nature, it requires thousands if not millions of programs to generate training data in ML techniques, and the computing research community does not have such a collection of programs in hand. Particularly, traditional benchmarks are short in number, and source code hosting websites (e.g., GitHub) have abundant code snippets instead of executable programs. This project aims to bridge this gap by creating a database including a large number of executable programs. The main goal of this project is to collect/write programs that stretch both computing power and memory capability of modern computers, leveraging coding exercise websites such as LeetCode and CodeForces. This database will subsequently be used for program performance predictive model training and testing. Besides the main purpose mentioned above, this project will also help students sharp their programming skills and benefit their future careers in both academia and industry, giving the fact that these coding exercise websites are the de facto means to prepare technical interviews.

Disciplines: Applied Mathematics

Hosting Site:

Brookhaven National Laboratory (BNL)

Internship location: Upton, NY