Past Projects

Project Description:

The goal of this project is to address arising computational challenges in improving performance of Artificial Intelligence and Deep Learning applications on state-of-the-art supercomputers. Particular focus is on improving the performance of current optimization algorithms (e.g. Stochastic Gradient Descent, Adam) applied to train statistical models (e.g. neural networks). Standard optimization algorithms update the regression weights of Deep Learning models in a strongly sequential fashion which is the consequence of the data batches successively updating the regression weights of the predictive models. This sequentiality in handling different data batches causes significant bottlenecks for the parallelization. Thus, current algorithms are extremely inefficient and computationally involved when statistical models are deployed and trained on high-performance-computing architectures. Although attempts to overcome this computational barriers are already underway, the improvements obtained to date are incomplete and limited in the scope of applications. Our project propose to develop general-purpose communication-avoiding strategies that can improve the scalability for the training of Deep Learning models without compromising convergence rates with respect to standard approaches.

The project requires dual commitment. On one hand, the implementation of the communication-avoiding techniques is planned. On the other hand, fundamental theory development is envisioned with the goal of ensuring that the reduction of communication and increase of concurrency does not deteriorate (and possibly improves) the convergence rates. To this end, the prospective student for this project will have the opportunity to combine mathematical studies and numerical experiments on benchmark problems and provide a proof-of-principle analysis of the algorithms.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

U.S. Citizenship is a requirement for this internship

Project Description:

In the Gulf of Mexico (GOM) and similar large bodies of water surrounded by land, seasonal warming in the summer months induces thermal expansion of the water surface through baroclinic processes. This seasonal variability is prevalent in the tidal signal at most NOAA tidal gages in the GOM region. For storm surge inundation studies, the timing of the tidal signal (flow or ebb stage) can contribute significantly to the overall surge inundation at the time of hurricane landfall. Traditional tidal analysis methods typically account for only the basic tidal signal structure, and tend to underestimate (or overestimate) the magnitude of the daily high (or low) tide. Current solutions in numerical modeling applications involve applying a static water level adjustment to the boundary conditions of the model (e.g., ADCIRC) to account for the variability in circulation. We seek a more universal approach to adjust the tidal signal for storm surge predictions. This project aims to characterize and quantify the seasonal variability in the tidal signal such that variability can be detected and corrected for storm surge numerical modeling applications. The findings of this project are expected to inform continued development of a probabilistic framework for detecting and correcting seasonal variability parallel to traditional tidal analysis methods in a follow-on opportunity. The prospective student intern will engage in research with mentors to develop an understanding of tidal analysis methods and processes. The intern will assist with determining the best methods/techniques to characterize and quantify the seasonal variability in the GOM region. If successful and/or if time permits, the intern will follow through in the detection and correction methodology. The intern will participate in weekly meetings with mentors to discuss progress, as well as document methodology and showcase results. The intern should have experience with time series analysis, scripting in high-level languages (e.g., MATLAB, Python), probability and statistical analyses methods, stochastic modeling, as well as interest in coastal water waves and storm/surge modeling.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

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

Internship location: Vicksburg, MS

Project Description:

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

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Disciplines: Applied Mathematics, and Mathematics (General)

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Disciplines: Applied Mathematics, and Mathematics (General)

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Disciplines: Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Disciplines: Applied Mathematics, Biometrics and Biostatistics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Disciplines: Applied Mathematics, and Biometrics and Biostatistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

We are building a generic radionuclide transport model of the vadose zone of the northern Negev region to develop capabilities required to evaluate potential vadose zone sites for nuclear waste disposal in Israel. This model will demonstrate the applicability of the Negev vadose zone for nuclear waste disposal, highlight potential limitations and data gaps, and refined at a later date for a specific site or expanded to include deeper geologic layers. The work plan combines experimental and numerical work. It is built in a structured manner and organized into three phases.

In Phase 1 – Year 1, a conceptual model has been be built by collecting all the available data on the geology, geohydrology, geophysics and geochemistry of the Negev area. The conceptual model serves as the basis for building the Geologic Framework Model (GFM) grid, used as the basis of the Negev subsurface Hydrogeologic and Flow and Transport process models.

In Phase 2 – Year 2, uranium batch sorption and column desorption experiments have been be conducted on rock samples of different lithologies initially using one radionuclide.

In Phase 3 – Year 3 of this study (2020), the radionuclide transport model will be completed, as well as the Negev subsurface hydrogeologic model. These will be integrated into a prototype flow and transport model for the Negev potential repository site. Preliminary site recommendations for the Negev area will be provided to the IAEC based on findings, to direct future siting decisions on potential vadose zone waste disposal sites. This project will focus on significant geostatistical analysis and creation of a complex 3-D simulation of radionuclide transport.

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

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Combinatorial algorithms are indispensable in factorization-based sparse linear equations solvers. Examples include reordering of a sparse matrix to limit the amount of fill in the factorization and scheduling tasks in parallel factorization. Finding an ordering that minimizes fill, and hence memory usage, of the solver is an NP-complete problem. Finding an optimal scheduling is also NP-complete. Over the past decades, high-quality heuristics have been developed for finding good approximations to some of these combinatorial problems. Traditionally, the development of these heuristics has focused on the quality of the solution. However, due to the ever-increasing degree of parallelism, they are becoming serious bottlenecks. Although efficient sequential implementations of the heuristics exist, these techniques are often hard to scale to multiple processing nodes. Instead of trying to come up with new, more scalable heuristics that achieve the same quality, it is time to explore radically new, out-of-the-box approaches. Deep reinforcement learning (DRL) is such a technique, which promises much more scalable solution.

The ability of DRL to learn directly on rules to discover new policies is of particular importance here. The very nature of NP-complete problems makes training a deep learning algorithm intractable in a supervised way. Without a brute force approach, one cannot compute the optimal solution of a problem, making it impossible to fabricate input-output data to train the algorithm. Training with data constructed from known heuristics will limit the achievable quality. Hence, enhancing the policies discovered by the algorithm, corresponding to the reinforcement part of the approach, is crucial. In this project, the candidate will investigate the use of various training reinforcement techniques for several combinatorial problems arising in sparse matrix factorizations, as well as explore configurations of the deep neural network.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

The project is to develop an autotuning software framework via statistical and machine learning techniques, such as multitask and transfer learning using Gaussian process. The goal of this work is to help the HPC codes (including parallel mathematical libraries and simulation codes) to choose the near-optimal parameters setting on a large-scale parallel machine, which take into account the characteristics of the input problems. The typical minimization metrics are runtime and memory usage. Since each execution (“function evaluation”) of the HPC code is expensive and takes a lot of resources, it is not feasible to use a brute-force approach (e.g., grid-search) to search for optimal parameters. Therefore, it is critical to “learn” some knowledge from the limited number of executions with certain input instances and build a prediction model for the unseen tasks.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

In many scientific applications, computer simulations are used to approximate complex physical phenomena. These simulations usually have parameters that must be adjusted in order to obtain the most accurate simulations. Accuracy is assessed by comparing the simulation output to observation data. However, these data are often noisy, and therefore parameter inference is needed to determine those simulation parameters that most likely explain the observations.

Simulations are often computationally expensive and may require several minutes or hours per run. Thus, during inference, we cannot query the simulation model thousands of times in order to find the desired parameter posterior distributions. Moreover, simulations are often provided as black boxes, i.e., there is no analytic description available and inference methods that are based on adaptive exploration of the sample space are needed. Previously, methods have been developed that exploit Gaussian process models as surrogates of the expensive simulation in Bayesian inference. However, these methods do not scale well with an increasing number of sample points and parameters.

In this project, your research will focus on the development of scalable inference algorithms that are efficient and effective for computationally expensive models. In order to achieve this, your work will involve the development of new sampling strategies that adaptively explore the potentially large-dimensional parameter space; the use of dimension reduction and sample space reduction methods; the use of Gaussian process models (or other types of surrogate models); and Bayesian inference methods. You will develop a suite of fast-to-compute test problems to assess the performance of your developed algorithm and finally apply it to a real-world science problem.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

This project explores the use of Probabilistic Graphical Models (PGM) such as Markov Random Fields (MRF) along with deep learning models to tackle image analysis problems at scale. Those two frameworks have been widely used individually with success in the area of image processing. Recent works propose the combination of PGM with Deep Neural Networks allowing: (1) easier and more efficient PGM optimization (2) incorporating learning into the PGM. Research scope will include not only the mathematical modeling behind these approaches, but also code development taking into account optimization for large datasets. Research and development will be applied to datasets obtained from state of the art instruments in one or more fields such as material science, medicine, biology, chemistry, others. Accepted applicants will collaborate directly with Dr. Perciano, who is a Research Scientist with broad expertise in image processing and analysis, computational statistics, high performance computing and machine learning. Interns will also network with and will have the opportunity to collaborate with members of the Data Analytics and Visualization group at LBNL and the Computational Biosciences Group.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Accurate models for the time evolution of stellar matter undergoing nuclear reactions are critical for simulating a variety of astrophysical problems spanning a large dynamic range in space and time. These include nuclear burning in turbulent convection, thermonuclear supernovae, core-collapse supernovae, X-ray bursts, and many other explosive events. The nucleosynthesis of a reacting volume of fluid can be described by a set of stiff, coupled ordinary differential equations (ODEs) in time that can be expensive to integrate and generally require implicit methods.

This project will apply machine learning methods to approximate the time integration of ODEs describing nuclear reactions as an alternative to integrating the reaction ODEs in hydrodynamics simulations. Our ultimate goal is to eliminate the computational expense of in-situ implicit integration for nuclear reaction ODEs entirely while preserving underlying physics including energy, baryon number, and lepton number conservation.

In this project we will first explore the size and topology requirements for neural networks to reproduce the nucleosynthesis and energy generation from nuclear reactions given the physical constraints above. Your research will determine the optimum neural network configurations for representing a range of different nuclear reaction systems applicable to open problems in astrophysics. We will generate training data sets from existing tools for implicit time integration of these systems of reactions across a range of representative thermodynamic conditions. You will also collaborate with astrophysicists to verify your neural network models in astrophysical burning scenarios where reactions and hydrodynamics are strongly coupled.

You will work with an interdisciplinary team of applied mathematicians, engineers, and astrophysicists in the Computational Research Division at Lawrence Berkeley National Laboratory.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Scientific machine learning (SciML) is a new and rapidly evolving field of research, lying at the intersection of machine learning and scientific computation. SciML focuses on how to incorporate the success of data-driven machine learning models to enhance physics-based simulations in computational science and engineering applications. In order to apply these methods to large-scale problems, efficient and scalable algorithms for high-performance computing platforms are required.

Residual networks, a powerful tool often used in deep learning applications, can be interpreted as discretized versions of dynamical systems. This interpretation can lead to new insights about the behavior of these networks, and inform the design of training algorithms.

This project will focus on the application of recent advances in parallel-in-time integration and optimal control to develop training algorithms that are scalable and parallelizable across the layers of the network. Additionally, a goal of this project is to develop novel discretizations for the weights of the network based on spline bases, allowing for deeper networks without increasing the complexity of the corresponding optimization problem. These algorithms will be run on large-scale, massively parallel supercomputers, and will be applied to relevant SciML problems.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Livermore National Laboratory

Internship location: Livermore, CA

Project Description:

This project develops an efficient solver for the quantum dynamical control problem, aiming at 1 M degrees of freedom (DOF), corresponding to a 20-qubit system. The need for optimal control of quantum systems lies at the heart of several emerging technologies from quantum sensing to dynamical control of chemical reactions and, most significantly, quantum computing. Solving the quantum control problem amounts to finding time-dependent control signals that are applied to the quantum computing hardware to guide the quantum states through a prescribed sequence of quantum logical operations. Finding the control signals can be cast into an optimization problem, under the constraints imposed by the time dependent Schroedinger equation, which governs the dynamics of the quantum system. Precise, fast and energy efficient control signals are required to realize the quantum logical gates that constitute the building block for quantum algorithms and to initialize a quantum system from a thermal state. Additional control signals are needed for extracting the information stored in the quantum states at the end of the simulation. In order to meet the demand for this capability we are developing a gradient based optimization algorithm, utilizing the adjoint Schroedinger equation to effectively compute the gradient of the objective function, implemented on a high-performance (classical) computing platform.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Livermore National Laboratory

Internship location: Livermore, CA

Project Description:

The goal of this project is to develop tools for image and shape analysis, by leveraging scientific computing and machine learning algorithms. Various research opportunities exist in the following topics:
- Shape spaces, shape analysis and statistics,
- Image segmentation, region and geometry detection in 2d/3d images,
- 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

Internship location: Gaithersburg, MD

U.S. Citizenship is a requirement for this internship

Project Description:

Vehicles rely on constant communication of many electronic control units (ECUs), little computer, which broadcast messages across a few controller area networks (CANs). Various ports and indirect communications (E.g., Bluetooth, wifi, etc.) have exposed this critical in-vehicle network to cyber attacks, e.g., the well-advertised Jeep hack, stopping a vehicle remotely while it was driving on a highway. Exacerbating the problem for defensive research is that there is no available translation of the CAN bus packet contents (bits) to the vehicle’s functions (e.g., speed, rpms, brake lights, …), and every model has different encodings. Basically, we can see all the messages, but we do not know what they mean, and it varies per model. The goal of this project is to use data science to aid in understanding and defending the vehicle network communications. We are currently working with regression techniques to reverse engineer signals in the data, and manifold learning and deep learning techniques for building anomaly detectors. We collect and test results on real cars and strive to implement detection capabilities in hardware. This internship seeks folks interested in learning and implementing algorithms to test detection accuracy.

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

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

Until fault tolerant quantum computers are readily available efficient program encodings minimizing circuit depth and associated errors are needed. This project focuses on the decomposition of a wide variety of Hamiltonian simulation and information processing unitaries into a set of operations physically implementable by many-body analog evolution with transmon-based superconducting qubits. The work will consist of employing unitary decomposition methods to compile hardware-efficient quantum programs.

Disciplines: Applied Mathematics, Geometry, Mathematics (General), Probability and Statistics, and Topology

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The Surveillance, Epidemiology, and End Results (SEER) Program of the National Cancer Institute provides structured and unstructured data from population-based cancer registries across multiple states. The SEER database provides information about demographics, cancer site, treatment, and follow up status. Leveraging the computing resources at Oak Ridge National Laboratory, the goal of this project is to analyze the SEER data to predict long term cancer outcomes based on socioeconomic status. With unsupervised learning methods, we will first create visual tools (ex. graphical networks with clustering algorithms) to broadly understand the SEER data. From there, we will leverage artificial intelligence and machine learning to develop a predictive model that utilizes multiple variables from the SEER registry. The ultimate goal is to understand which socioeconomic factors are key contributors to the treatment and follow up status of patients. This internship seeks applicants with experience in programming (ideally R or Python), and an interest in applying a multitude of mathematical and statistical skills to solve key problems in health care.

Disciplines: Applied Mathematics, Biometrics and Biostatistics, and Mathematics (General)

Hosting Site:

Oak Ridge National Laboratory

Internship location: Hanover, NH

Project Description:

Phase Field Modeling (PFM) is a technique used to track interfaces using a “phase-field” that takes, for instance, a value of 0 on one side of the interface, a value of 1 on the other side, and smoothly varies between the two through the interface. The interface can then be defined by the iso-surface where the function takes the value 0.5. This technique is used a lot for modeling solidification in metals, where the phase variable would take the value “0” would correspond to  “liquid” and the value “1” to “solid”. A more specific application of recent interest is modeling solidification in metallic alloys during the process of additive manufacturing. In practice, PFM leads to solving a system of coupled partial differential equations to calculate the time evolution of the interface/phase-field, the composition of the alloy, and possibly a coupled temperature field. These equations can be discretized in time using an implicit scheme (backward Euler) and in space using a finite-volume approach. These discretized equations can then be solved using a Jacobian Free Newton Krylov (JFNK) approach. Solving the resulting linear systems iteratively --- by GMRES for example --- requires a good preconditioner. Geometric multigrid works well in practice to precondition the diffusion equations to be solved for composition and temperature. In this project, we want to extend the general technique described above to the context of a moving “frame”, that is a computational domain that follows the solidification front and moves at the velocity of the interface. Doing that adds a “convection” term to the equations, with a velocity given by the velocity of the moving frame. This has been done before for explicit time discretizations, but not for an implicit time-stepping. The difficulty in an implicit approach is to find a good preconditioner that can handle well the diffusion part and the convection part of the operators. The idea to be developed in this project is to use a preconditioner based on an operator splitting idea. This research will require some coding in an open source C++ code developed by mentor and others, which uses the Sundials package for time-integration. This research is likely to lead to a peer-review publication.

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The goal of this project is to develop hybrid algorithms for the numerical simulation of complex particles systems. These algorithms combine fluid and kinetic models in order to construct highly efficient simulations that incorporate non-equilibrium kinetic effects only when necessary for simulation accuracy. In this project, the student intern will develop numerical methods, perform numerical analysis, and implement methods numerically using modern software tools.

Disciplines: Applied Mathematics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The purpose of this project is to predict human mobility patterns following disasters due to failure of critical infrastructures, specifically, transportation networks. Understanding human mobility pattern following a disaster event, such as a tropical storm, is crucial for assessing impacted and displaced population. Given that many communities lack information about how the failure of an infrastructure following a disaster can impact emergency management efforts, it is crucial to understand how failure of critical infrastructures, specifically, roads impact mobility pattern of impacted populations and subsequent origin and destination locations. Given the availability of large volume of heterogeneous location and mobility data, this project focuses on: (i) understanding human mobility pattern during normal conditions using heterogeneous big data (unstructured - social media and other crowdsourced textual and imagery data, and structured geospatial data - infrastructure and disaster data) using activity based intelligence, network analysis and trajectory data mining, (ii) developing reinforcement learning based recommender system to predict mobility pattern during and following a disaster by accounting for damaged and unusable road networks, and (iii) uncertainty quantification of the outputs.

Other than being transformative, the scientific impacts of this research include (i) deriving fundamental understanding about human mobility behavior that could be used to develop strategies for humanitarian response and infrastructure planning, (ii) deriving insights into the interaction between human mobility and infrastructure network (i.e., roads) that is pivotal for infrastructure failure planning, restoration and emergency response. The broader impact of this project is the creation of next-generation response tools for first responders, decision-makers and stakeholders to aid with decision-making and preparedness activities. We have recently expanded to implementation of activity-based intelligence (ABI) to derive origin and destination matrix that would be used in the reinforcement learner. The ABI will be evaluated in near real-time by accounting for changes in transportation network access and extreme event impact areas such that different sets of origin, destination and network will be determined used the reinforcement learner under different circumstances.

Disciplines: Analysis, Applied Mathematics, and Operations Research

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The Oak Ridge National Laboratory has many parallel computing systems as well as many R language parallel computing tools developed by the pbdR project (pbdr.org). The student’s project will involve either using these tools to perform various statistical analyses on a mutually agreed on large data set or developing more parallel statistical computing tools. While some specific project topics are available (such as parallelizing knockoffs or polynomial regression tools), it is also possible for the graduate student to further own thesis related research with parallel computing on large parallel systems. Ideally, the student will already have considerable experience with R and possibly some exposure to parallel computing.

Disciplines: Biometrics and Biostatistics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

Peridynamics is a nonlocal reformulation of classical continuum mechanics suitable for material failure and damage simulation. Peridynamic models have been applied to simulate a variety of engineering problems, particularly those involving large deformation and crack propagation. This project will study various numerical methods to advance peridynamic capabilities, in terms of simulation accuracy and efficiency.

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

Atomistic models have been shown to be effective for modeling and simulation of many materials science problems. One of their main drawbacks is a high computational expense, which limits their application to very small systems. Atomistic-to-continuum (AtC) coupling is a multiscale modeling technique to attain accurate representation of atomistic phenomena in large-scale systems. This is achieved by employing atomistic models only in small critical regions, while coupling those to continuum formulations. Quasicontinuum is a class of AtC coupling methods. This project will investigate a novel high-order quasicontinuum approach based on a blending methodology and study numerically and analytically its performance in AtC coupling problems.

Disciplines: Applied Mathematics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The aim of this project is to identify materials that retain their strength for tens of thousands of hours during service at temperatures in excess of 650 C. We will gather relevant alloy data, assess data quality, and use data analytics and machine learning to identify key factors governing alloy performance. Advanced data computing and data science methods have emerged over the past decade and have the potential to transform the energy-materials sector.  There is a need for integrated data solutions, tools and databases, to support materials data analytics to meet end user needs.  At present, materials data and information persist in largely disparate and differing forms.  There is a need for new data science driven tools to help find, acquire, and transform these existing datasets and put them to work for energy-materials research.  Building this data foundation and developing tools and algorithms that will help improve efficiency of data acquisition and transformation is integral to meeting this need.  For this effort, a solid foundation of data and information is necessary to understand the types of information about key materials that are currently collected, where that data and information exists presently, and define an approach for transforming data from a range of sources to meet end user needs.  This summer, the team will focus on selecting 1 or 2 key materials as a use case to address the needs above, develop a work flow for addressing those needs, and initiating development of cohesive database and data analytics for those key materials where data resources acquired can be integrated.

Disciplines: Analysis, Applied Mathematics, and Probability and Statistics

Hosting Site:

Pacific Northwest National Laboratory

Internship location: Richland, WA

Project Description:

In this project we continue an ongoing effort focused on the development of a unified nonlocal theory. The ultimate goal is to derive a universal theory of nonlocal operators that spans a broad spectrum of nonlocal processes and has, as special instances, the classical calculus and the fractional calculus. The foundation of such unified calculus is the well-established nonlocal vector calculus for "truncated nonlocal operators". This new theoretical framework would provide the groundwork for new-model discovery and, hence, enable modeling and simulation of intrinsically nonlocal phenomena that have not been studied due to the lack of theory. In this project, we focus on necessary preliminary results: the establishment of equivalences of unified nonlocal operators and common nonlocal operators used in mechanics.

Disciplines: Applied Mathematics, and Mathematics (General)

Hosting Site:

Sandia National Laboratories

Internship location: Albuquerque, NM

U.S. Citizenship is a requirement for this internship

Project Description:

Goal of this project is to generate series of anisotropic tetmeshes for solution convergence. Initial tetmesh is generated by using Sandia’s geometry and meshing toolkit Cubit. Cubit uses Delaunay based algorithm with geometrically guided controls to generate an initial tetmesh. First step in this project is the extract a tensor field that captures the initial mesh characteristics. The tensor field defines tetrahedral elements’ size, shape and orientation throughout the domain. Next step is to scale the metric tensor field so that desired number of output elements can be obtained. The scaling factor is defined as the ratio of number of elements in the output mesh to number of elements in the initial mesh. This scaling factor is provided by the end user. While the anisotropy of the mesh elements is controlled globally through the applied scaling measure, the resulting volumetric constraint is controlled through the necessary local modifications of the mesh at each iteration of the mesh generation procedure. Such local modifications are contingent upon the notion of the metrics defined at discrete points. In general, the metrics act as controls for the size, shape, and orientation of the generated meshes. We rely on the theoretical foundation of the log-euclidean framework to generate metrics at nodes of a discretized mesh. This framework establishes a one-to-one relation between the vector space of symmetric matrices and the vector space of tensors on the computational domain. Nodal metrics are expressed in the form of a weighted geometric mean of the mesh element metrics. Open source Omega-H library will be used to adapt the initial mesh to match the scaled tensor field defined at the nodes in generating output anisotropic tetmesh.

Student will work closely with developers of Cubit and Omega-H. As most of the infrastructure is already in place, there is a good possibility to publish the work in a mesh generation related international conference. One of the goals of this internship is to get results from this continued project and potentially publish this research in the International Meshing Roundtable conference.

Disciplines: Applied Mathematics, and Geometry

Hosting Site:

Sandia National Laboratories

Internship location: Albuquerque, NM

U.S. Citizenship is a requirement for this internship

Project Description:

The main focus of this project is to utilize low Size, Weight and Power (SWAP) sensors to map and survey building interiors and subterranean environments. One key aspect of this project is sensor fusion leveraging the Robot Operating System (ROS). Another is exploring Simultaneous localization and mapping (SLAM). The optimal configuration (hardware and software) will then be incorporated onto a GVR-bot UGV to determine the accuracy of the resulting point cloud.

Disciplines: Geometry

Hosting Site:

U.S. Army Corps of Engineers, Geospatial Research Laboratory

Internship location: Various Locations

U.S. Citizenship is a requirement for this internship

Project Description:

Photon-counting LIDAR systems collect high-resolution 3D data from high altitudes across large areas through use of low signal-to-noise (SNR) receivers.  These devices have larger amounts of noise versus traditional, low-altitude LIDAR sensors.  The two primary noise sources can be characterized as uncorrelated Poisson noise and crosstalk noise correlated with stronger, real detections off of objects in the scene.  This project will apply new, untested mathematical methods for de-noising and signal processing to raw photon-counting LIDAR datasets.  Participant will collaborate closely with the primary signal processing algorithm developer and apply their knowledge of mathematical methods to the problem through algorithm development in either Matlab or Python.

Hosting Site:

U.S. Army Corps of Engineers, Geospatial Research Laboratory

Internship location: Hanover, NH

Project Description:

Research Objective: What biological factors drive the basic individual interactions in bird flocks or fish schools, causing them to coalesce into larger groups that move fluidly in unison?  More importantly, how can such factors enable the contagious spread of an idea or action? We aim to advance our ability to forecast critical transitions by unraveling the building blocks of social contagion in a model system using recent advances in information theory. Evidence strongly suggests that animals may adopt a topological interaction mechanism as a means of coordinating their activity, but is drawn from a statistical mechanics approach that largely ignores the biological origins of the phenomena. Recent discoveries suggest that animal attention to physical cues is a crucial factor in driving collective action, thereby suggesting a flexible biological mechanism which parades as a fixed topological pattern. Hypotheses for the interaction mechanisms driving collective action have been traditionally validated using computer simulation to match group-level consequences with qualitative features observed in wild populations. However, there has never before been a means to more directly distinguish which, if any, of these hypothetical mechanisms are used by real animals to self-organize, and empirical evidence to decisively settle the issue is lacking. To address this problem, we propose a novel test of the Weber’s Law perception mechanism—i.e., that perception is driven by a fold-change in the magnitude of sudden velocity fluctuations—using social behavior in fish as a prototype for collective animal motion with state-of-the-art information theory metrics that can distinguish between direct influence (A→B) and indirect influence (A→C→B). More specifically, we hypothesize that if animal collective motion is driven by a Weber’s Law selective attention mechanism, then individual fish will always be most directly influenced by local velocity fluctuations that exceed the group average.  To test this hypothesis, we will use computer simulation to understand how influence from velocity fluctuations can be inferred from group-level data using information theory metrics that distinguish causation from correlation. We will then determine whether velocity fluctuations drive individual attention in fish subjects experimentally using decision trials. We will manipulate key physical features (size, speed, direction) of virtual stimuli and track the reaction of our subjects, both in isolation and in groups. Experiments will determine whether individual dependence on velocity fluctuations propagate beyond pairwise interactions or three-body interactions to drive schooling at the group level.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

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

Internship location: Vicksburg, MS

Project Description:

We plan to develop a robust machine learning modeling framework to relate biomass properties variability at tissue levels by rapid NIR spectroscopy linked hyperspectral imaging (HSI) system with feedstock handling and conversion performances and to optimize conversion-ready designer feedstock for a biorefinery. Geospatially sourced corn stover (CS) - hydrolysis target, and southern pine forest residues (SPFS) - pyrolysis target, from various locations will be sampled, characterized, labelled, and screened into various tissue components corn stover5: cobs, leaves, husk, stalks/internode, others; pine forest residues: needles, bark, juvenile wood, chips/branches, others). When a tissue component does not offer improved conversion yield, other conversion pathways such as gasification, combustion or production of biochemical conversion will be explored. For example, juvenile wood contains more than 60% of hemicellulose, which could be an excellent feedstock for producing xylitol rather than pyrolysis. However, the economics of separating and producing juvenile wood should be considered and it will be considered in this proposal. The physical and chemical properties of each tissue component will be determined analytically while developing rapid measurement method by Hyperspectral imaging and sensing techniques to collect quick properties data. The performance metrics for mechanical screening equipment for tissue fractions (e.g. energy use, yield), powder handling characteristics after grinding, and conversion potentials (e.g. yield) of each tissue fractions will be determined at lab-scale. The tissue properties data will be used to train, validate and test machine learning based predictive models such as Artificial Neural Networks (ANN) with high correlation to feedstock performance metrics. The predictive models can serve as a decision support tool to identify, and/or design and optimize uniform conversion ready tissue fractions at low costs. The intern will collaborate with a postdoc research fellow in reviewing and finding the best artificial neural network (ANN) model to correlate biomass chemical compositions with conversion performances for biorefineries.

Disciplines: Analysis, and Biometrics and Biostatistics

Hosting Site:

USDA Forest Service Forest Products Laboratory

Internship location: Madison, WI

Project Description:

1. The effects of non-response can degrade data quality.
2. This method of survey data collection is essentially a form of cluster sampling with unequal cluster sizes.
3. If N, the number of clusters, is not known but can be estimated reasonably well, the above MSE formula suggests it will have only a minor impact of the results.

Disciplines: Analysis

Hosting Site:

USDA Forest Service Forest Products Laboratory

Internship location: Madison, WI

Project Description:

Image compression is a very active field of study, with new methods being constantly generated. The need for improvements in image compression quality is growing in the field of HPC simulations because of the exponential trend in data generation. There exists an untapped potential in this situation due to the nature of simulated data that is not currently exploited. Simulation data from numerical systems of partial differential equations exist on a solution manifold. Thus, the manifold hypothesis in machine learning---which states that real-world, high-dimensional data lie on low-dimensional manifolds embedded within the high-dimensional space---is concrete for simulation data. We can therefore expect that identifying this map to the low-dimensional manifold will provide ideal compression for HPC Simulations. This project will focus on designing ideal compression for HPC simulation.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

U.S. Citizenship is a requirement for this internship

Project Description:

For an electromagnetic pulse (EMP) application, the MCNP code is used for photon and neutron transport in air and to compute the energy deposition rate and photocurrent density. MCNP estimates these quantities as a function of space and time. The MCNP calculations are time consuming and can take weeks to complete. This project seeks to build and train deep neural networks to estimate the energy deposition rate and photocurrent density as a function of space, time, source particle type (neutron or photon), source particle energy, and source height. The neural networks will be trained on MCNP results for photon and neutron transport in the atmosphere. The students will focus on  building and training the neural networks and not on running MCNP to construct the training and  testing data set. However, the students will be introduced to the topics of MCNP and EMP, and will learn to run MCNP for a few problems.

Specific activities on the project include:

1. Generate training and testing data for machine learning predictors using the existing set of 1500 MCNP runs.
2. Build machine learning predictors for the energy deposition rate and electron current density for photon and neutron sources.
3. Build physics-based regression models that provide an intuitive understanding and compare the results with general machine learning approaches.
4. Learn to use MCNP and an understanding of EMP.
5. Analysis of MCNP and regression results. Python is currently used and this is the preferred language.

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

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

We are developing efficient physics-informed neural network reduced order models (NNROMs) to accelerate complicated, large-scale physical simulations. Our current LLNL-developed 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 will extend our 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 turbulence problem by training an autoencoder neural network on 2D turbulence model data and implementing NNROM on the turbulence model. Depending on the results, we will write a journal paper together. Our NNROM will be general enough that by the end of summer, the student will be able to apply the NNROM 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

Internship location: Livermore, CA

Project Description:

This project focuses on the development of new algorithms for preconditioning high-order finite element operators without explicit matrix assembly.

Emerging high-performance computing architectures make high-order finite elements more attractive, because high local arithmetic complexity often allows them to deliver higher accuracy for a similar compute time compared to low-order methods. Efficient implementations of the operator-vector multiply in this context are very fast with a matrix-free implementation, but matrix assembly is not practical at high polynomial order.

The lack of an assembled matrix poses a challenge for solvers, where the standard workhorse of algebraic multigrid depends intimately on such a matrix. The goal of this project is to extend the general, automatic nature of algebraic multigrid to a setting where no explicit matrix is available.

The project will involve discovering new algorithms for preconditioning and solving linear systems in this context, analyzing their theoretical convergence properties and computational costs, and implementing them in practical software packages.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Livermore National Laboratory

Internship location: Livermore, CA

Project Description:

Description:
Many of the leading proposals for obtaining a quantum advantage on near-term quantum computational hardware are based on known hardness results for the task of sampling from the output distribution of low-depth quantum circuits. However, while the task of sampling the output of these circuits is known to be hard when performed at sufficiently high precision, it has also been shown that, in some cases, there are efficient classical algorithms for computing output probabilities of these circuits to inverse polynomial precision. These algorithmic results, while currently limited, call into question whether the near-term quantum circuits currently used to perform hard sampling tasks can ever be used to solve hard *decision* problems. In this project, in an attempt to make progress on this fundamental question, we will focus on notable classes of near-term quantum circuits for which this question is unresolved, and seek to design algorithms to efficiently estimate their outputs. Designing these algorithms is an opportunity to develop and put to use a strong intuition for both quantum computation and classical algorithms.

Preferred Prerequisites:
A first course in quantum computing or an understanding of the fundamentals of gate-model quantum computation, A proof-based classical algorithms course or equivalent knowledge, Probability, Linear Algebra, Mathematical Maturity.

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

Hosting Site:

National Institute of Standards and Technology

Internship location: Gaithersburg, MD

Project Description:

This project will collect, curate and manage data and information from the literature, processing and characterization experiments, and multiscale simulations. The database by itself will offer enduring value by bringing hidden data to light, preserving it, and making it available for future research. In addition, data analytics will connect the results of simulations and experiments to achieve new scientific understanding of solid phase processing (SPP) of alloys.

Recent advances in data science offer an exciting opportunity to advance the science of SPP by identifying correlations in the large volume of data generated during processing and defining the key features that control the microstructure of alloys. Data management and data analytics integrated with physics-based simulations, validated using data from experimental processing, process-scale characterization and advanced in situ and ex situ characterization, are needed to optimize the processing conditions. Connecting the SPP parameters to microstructural evolution and phase stability is a daunting challenge. Data analysis will identify the key processing parameters, out of many, that control microstructural changes. The expense and time of generating data from processing runs and beamline experiments makes it vital to collect and curate the data. In addition, data analysis will help link the physics-based models at different scales and develop reduced-order models of alloy performance. This project will establish a data management framework and data analytics tools to optimize processing and integrate the strengths of the experimental and modeling efforts.

Disciplines: Analysis, Applied Mathematics, and Probability and Statistics

Hosting Site:

Pacific Northwest National Laboratory

Internship location: Richland, WA

Project Description:

Using reduced precision floating operations (single or half-precision instead of double precision) can speedup significantly computer simulations on modern computer architectures, especially for GPU-accelerated nodes. On the other hand, reducing precision can affect the quantitative results of a simulation in a way that is not acceptable. In this project, we will explore ways of taking advantage of mixed-precision algorithms in electronic structure calculations without affecting the physical results of our simulations. Since solvers can be quite complicated to analyze globally and very little research has been done with mixed-precision in that domain, we will focus on simplified problems that involve only a subset of the unknowns of the real problems. We will also examine how the lessons learned from these simplified problems carry to a complete solver on real-applications.

This project will require modifying, adding functionalities, and running an existing open-source C++ code. Results may lead to a publication in a peer-reviewed scientific journal.

Disciplines: Applied Mathematics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

We are interested in exploring the possibility of leveraging the multilayer structure that is used in neural networks to improve other learning methods. Particularly, we are exploring how we can create multilayer multivariate adaptive regression splines (multilayer MARS) that can be used to make predictive models with less complexity in depth at the expense of more complexity in each layer, though perhaps with fewer total parameters. We intend to use these models for building surrogates for complex computer models.

Disciplines: Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Adversarial attacks on neural networks refer to the ability of certain inputs to “trick” a neural network into giving an undesired output. Widely circulated examples include image classifiers that can be fooled into significant mis-classifications by seemingly modest perturbations of an image from the training data set. The goal of this project is to explore geometric properties of the data that may help identify what makes an effective adversarial attack. While much of the literature in this area looks at the problem from a statistical perspective, this project will focus more on the geometric perspective, such as the importance of sampling density and the Euclidean distance-to-hull of an adversarial input. Project tasks will include both mathematical analyses and numerical experimentation, using the extensive computational resources of HPC systems at LLNL. Results from this project have the potential to provide new insight into generative adversarial network design, a field of growing importance in scientific machine learning.

This project will be performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. IM release number LLNL-ABS-816020.

Disciplines: Applied Mathematics, Geometry, and Probability and Statistics

Hosting Site:

Lawrence Livermore National Laboratory

Internship location: Livermore, CA

U.S. Citizenship is a requirement for this internship

Project Description:

Drought indices based on modeled environmental data, a digital elevation model, and soil characteristics are broadly used to define the degree of hydrological deficits or agronomic drought. However, modeled environmental data are less accurate in high mountain ranges, and many of the soil characteristics are unknowable at any scale in this terrain. Approximately 35 million Californians rely on west- and east-side Sierra Nevadan water resources. The impact of the last two droughts (1999-2002; 2013-2016) were extreme, and the level of drought was unanticipated. Roughly 135 million trees died in the Sierra Nevada. Water management districts use drought indices and their trends to allocate water resources. We propose modeling environmental drivers of physiological pine drought stress using an existing, 20 year, on-site environmental and biological data that span the two extreme droughts to test the veracity of commonly used drought indices in the Sierra Nevada (USDM;SPEI12;SCPDSI;SPEI12;VegDri).

We propose two statistical modeling programs: to identify how tree physiology depends on local environmental covariates, and to test effectiveness of different drought indices in predicting physiological tree drought stress, the 'biological barometer.' Flexible additive models can be used to identify both linear and non-linear dependence of tree physiology on the environmental covariates, and to estimate the functional form of such responses in the non-linear case. For the classifier problem, machine learning methods (e.g., tree-based classification, kernel methods, and linear classifiers) can all be trained with our data, and their predictive power evaluated. Other potentially useful models include hidden Markov models, since the data is sequential, and underlying but hidden states may be driving the observable responses. The available data lends itself to multiple modeling approaches which the intern will develop and evaluate.

Disciplines: Applied Mathematics, Biometrics and Biostatistics, and Probability and Statistics

Hosting Site:

USDA Forest Service Pacific Northwest Research Station

Internship location: Corvallis, OR

Project Description:

The equations of gasdynamics consist of a hyperbolic systems of partial differential equations (PDEs) which are generally characterized by the presence and evolution of shocks. Their solution has been historically obtained using finite-volume (FV) schemes because of their ability to resolve steep solution gradients. However, despite their robustness, FV schemes require elaborated modifications to reach high-order accuracy and/or to handle complex geometries.

More recently, discontinuous Galerkin (dG) methods have become popular for the solution of PDEs because they offer a variational framework where variable order, complex-shaped elements are more naturally treated. High-order dG methods are very efficient in regions of smooth gradients but induce undesirable, and sometimes unacceptable, oscillations in presence of steep gradients and/or shocks. Therefore, dG methods are combined with suitably-defined limiting strategies when employed for the solution of hyperbolic equations.

During this internship, you will focus on the development of a numerical solver for hyperbolic PDEs that combines the advantages of FV and DG schemes and limit their shortcomings. More specifically, your research will involve the development of:
- a shock sensor used to identify regions where FV schemes should replace dG; and
- the coupling algorithm between dG elements and FV elements.

You will write the solver in AMReX (https://amrex-codes.github.io), a software framework for massively parallel, block-structured adaptive mesh refinement applications.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Device level modeling using unified simulation frameworks of complex energy systems has the potential to increase device efficiencies by enabling design optimization processes and technological innovations. Current modeling strategies rely on passive coupling between solvers at different length and time scales using parameterized representations. These approaches do not account for two-way coupling across the scales (e.g. transport affecting catalyst reactions), stochastic effects at the smaller scales (e.g. dendrite initiation in batteries), and state space exploration of unvalidated parametrized regions. NREL is currently exploring multi-resolution multi-algorithms frameworks for performing simulations of complex energy systems in the NREL mission space, including electro-catalysis devices, batteries, and biomass feedstock handling. The objective of this project is to develop high fidelity modeling frameworks that can leverage multi-resolution representations of the system to deploy algorithms tailored to the length scale under consideration and couple these algorithms in a unified framework. This project will involve studying scale coupling mechanisms for different mathematical models present in physics solvers to bridge scales (e.g. the atomistic-continuum scale), ranging from continuum solvers with varying physical assumptions to atomistic and particle-based solvers (e.g. DSMC, KMC, MD). Graduate students involved in this project will have opportunities for both publication and future collaboration.

Disciplines: Applied Mathematics

Hosting Site:

National Renewable Energy Laboratory

Internship location: Golden, CO

Project Description:

Recent observations indicate extensive permafrost thawing everywhere in the Arctic. Permafrost thaw triggers landcover changes, soil carbon release, soil subsidence, and adds to the increasing cost of infrastructure maintenance. Existing numerical models are unable to predict permafrost conditions effectively due to the scale restrictions or the computational burden. There is an urgent need for a new model that can represent the required spatial scale and be computationally effective. This project will require ability to quickly learn and apply numerical models to solve climate change related problems in the Arctic. Knowledge of the optimization theory, sensitivity analysis, and uncertainty quantification is a plus.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Pulsars are rapidly rotating, highly-magnetized neutron stars that were discovered more than half a century ago, yet, we do not understand the fundamental processes driving their electromagnetic radiation. Recent advances in computer architectures as well as numerical methods allow for detailed global simulations of the pulsar magnetospheres. WarpX (https://github.com/ECP-WarpX/WarpX.git) is an electrodynamics Particle-In-Cell code developed as part of the Department of Energy (DOE) funded Exascale Computing Project (ECP). The advanced numerical methodologies implemented in WarpX make it a unique computational tool to extend the state-of-the-art modeling techniques employed to investigate pulsar magnetospheres.

During the internship, you will focus on exploring mesh-refinement strategies to resolve the current sheet region in the pulsar magnetosphere required to study the critical magnetic reconnection phenomena using WarpX. In particular, you will analyze the growth-rate of the tearing-mode instability in the current-sheets formed in pulsar magnetospheres and compute the energy transfer from the electromagnetic fields to the kinetic energy of the particles. This project will further enable accurate predictions of the spin-down rate of the pulsar and comparison with observations for realistic systems as well as uncover the kinetic mechanisms driving the large-scale electromagnetic radiation.

You will collaborate with an interdisciplinary team of astrophysicists, applied mathematicians, computational scientists in the Center for Computational Sciences and Engineering (CCSE) as well as collaborate with plasma physicists in the Accelerator Technology and Applied Physics (ATAP) group at Lawrence Berkeley National Laboratory.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Complex computations typically require the interaction of a large number of subnetworks which must coordinate communication and computation. Intriguingly, considerable mounting evidence has shown that the brain can exchange information on an “as needed” basis and reconfigure computation “on the fly”. It is hypothesized that the prevalent oscillatory in the brain provide a substrate to flexibly coordinate computation.

We have shown (Kirst et al., Nature Communications, 2016, TEDx Talk 2019) that brain-inspired coupled oscillator networks can indeed dynamically coordinate information exchange. Using appropriate feedback they can be turned into self-modifying systems which effectively reprogram themselves (Kirst et al., COSB 2017). In this framework, information is flexibly routed and processed when encoded in fluctuations that ‘surf on top’ of intrinsic oscillatory dynamic reference states akin to how radio signals are broadcasted via amplitude or frequency modulations of electro-magnetic waves (AM or FM).

Building on our theory, we aim to develop a brain-inspired analog computing framework that employs collective network dynamics to coordinate large scale distributed computation and enable flexible and adaptive processing in dynamically self-reconfiguring neuronal networks.

These tools will have a broad range of applications, including dynamic scene understanding, attention guided computation, belief propagation based inference, as well as the coordination of large scale computation in ensembles of neuromorphic expert systems.

The project will expose the students to mechanisms for flexible computation combining analytical tools form information theory and stochastic dynamical systems (information dynamics), computational modeling as well as novel approaches to large-scale machine learning.

Disciplines: Applied Mathematics, Biometrics and Biostatistics, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Stochastic thermodynamics is an emerging framework for the description of the thermodynamics and statistical mechanics of stochastic systems far from equilibrium. In particular, the development of fluctuation relations provides analytical results for the statistics of thermodynamic path variables (such as work, heat, and entropy production) in the form of equalities, as opposed to the inequalities of the second law of thermodynamics. Fluctuation relations have generalized Landauer's principle to a variety of physical systems, setting a fundamental limit to information processing.

Our group has recently developed a quantum computing algorithm that uses a particular fluctuation relation, the Jarzynski equality, which relates the statistics of work generated during non-equilibrium trajectories to the equilibrium free energy (a thermodynamic function providing information about equilibrium phases). Our method utilizes real time Hamiltonian evolution to drive non-equilibrium protocols, and thus is naturally suited to quantum computing. We propose two directions building on our current method:

1. Determining optimal time evolution protocols, protocols that minimize dissipation along a quantum trajectory, and thus minimize the amount of information lost during a quantum process.
2. Extending this algorithm to sample other time extensive variables of quantum trajectories, whose rare statistics give information about quantum processes, quantum phase transitions, and error estimation; this advancement would lead to a more general quantum algorithm to measure large deviation functions, a task known to be exponentially costly for classical computing.

This project will involve a combination of analytical activities to develop the algorithms (familiarity with methods to sample rare events and measure complexity bounds could be useful) as well as implementing and demonstrating the algorithms on quantum hardware (some quantum computing experience is helpful but not required).

Disciplines: Mathematics (General)

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Although the fundamental framework of the nature or the laws of physics are well known, the ability to use them for predictive science remains to be limited, even with the most powerful computers today. Examples of such laws include Boltzmann, Maxwell and Schrodinger's equations. We are looking for a creative mathematician to join our team of interdisciplinary experts to explore the new possibilities of data science (especially DNN) constrained by physics considerations, and search for new insight and implications of these frameworks to plasmas, metamaterials, astrophysics, and quantum science.

Disciplines: Applied Mathematics, Geometry, and Mathematics (General)

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

"The Bayesian inference paradigm provides a probabilistic formulation for integrating information from complex models from observational or experimental data under uncertainty by updating the model parameters from their prior distribution to a posterior distribution. Solution to a Bayesian inverse problem involves the task of drawing samples from the posterior probability distribution to compute various statistics of quantities of interest. However, this is prohibitively expensive when the posterior distribution is high-dimensional; many conventional methods for Bayesian inference suffer from the curse of dimensionality, i.e., computational complexity grows exponentially or convergence deteriorates with increasing parameter dimension. This project will explore Machine Learning based alternatives to mitigate the prohibitive costs associated with solving Bayesian inverse problems."

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

Neutron stars are complex astrophysical systems where matter is compressed to extreme densities comparable to the interior of atomic nuclei. A variety of interesting physics may emerge under these conditions, including superfluidity arising from quantum-mechanical pairing between neutrons in the neutron star interior. We will investigate the astrophysical implications of neutron superfluidity by numerically solving the underlying partial differential equations (PDEs).

Superfluidity is a large-scale quantum behavior arising from particle pairing that macroscopically manifests as an irrotational fluid. Complex interactions between the superfluid and normal fluid components of a neutron star are thought to underlie mysterious phenomena known as “pulsar glitches,” observed as rapid changes in the angular speed of a spinning neutron star. We will investigate this physics by numerically solving the HBVK PDEs applied to normal and superfluid components in neutron star interiors.

With recent computational advances, we are poised to solve these PDEs with unprecedented fidelity, combining high order numerical discretization with adaptive mesh refinement algorithms running on supercomputers. In this project, we will work together to develop and test numerical simulations of superfluidity and you will also learn how PDE solvers are designed, implemented, and run on supercomputing systems.

You will collaborate with an interdisciplinary team of applied mathematicians, computational scientists, and physicists in the Computational Research Division at Lawrence Berkeley National Laboratory.

Disciplines: Applied Mathematics, and Mathematics (General)

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

The student will aid in the development of a new exascale-ready, multiscale software framework for physical modeling of electromagnetic signals with the flexibility for additional physics coupling, targeted at current and next-generation microelectronic devices. Examples include spin-based memory devices and quantum information processing circuits. The goal of this project is to reach into new energy-efficient application spaces with the incorporation of new physics, enabling improved design of next-gen devices. The student will leverage the extensive software and algorithmic expertise developed in collaboration with the Exascale Computing Project (ECP) Co-Design Center, AMReX, and related AMReX-based applications, WarpX. Specifically, the student will collaborate on (i) customization of existing algorithms and incorporation of new physics by adding in PDEs; (ii) update current implementation of boundary conditions, excitation, field evolution, etc., to accurately predict the newly added physical phenomena; (iii) explore leadership class GPU/multicore supercomputing architectures that will provide orders-of-magnitude speedup over existing capabilities. The outcome of this position include technical publications, conference presentations, etc.

Disciplines: Applied Mathematics, and Mathematics (General)

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Optimization is widely used to responsively determine the physical and financial exchange of wholesale electricity in organized markets. The established operations of power grid independent system operators (ISOs) are increasingly challenged by growing dependence on natural gas as a fuel and increasing penetration of uncontrollable renewable energy. This compels advanced methods for optimization-based gas pipeline operation and optimal allocation of reserve capacity to guarantee secure power and gas transmission function.

Such methods require robust optimization for the nonconvex, nonlinear, and spatiotemporally complex stochastic models of network flow physics under uncertainty, which presents mathematical and computational challenges. While formulations for uncertainty-aware power flow have been proposed, the use of dual solutions for price formation in the stochastic setting presents a fundamental conceptual gap. Yet such mechanisms, which apply optimization to synthesize physical models with business processes, are indispensable to incentivize adoption of advanced computational tools in practice. This project will develop a general formal mathematical setting for dual (sensitivity) analysis of nonlinear stochastic physical network flows modeled by partial differential equations on graphs, and create constructive optimization algorithms for computing the economic value of energy network load and production uncertainties and measures used to mitigate them.

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

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

U.S. Citizenship is a requirement for this internship

Project Description:

The prospective intern will engage in numerical modeling and mathematical analysis of the hydrodynamic problem at hand. In this project, the intern will have an opportunity to learn how to navigate through and utilize DoD’s HPC machines. He/she will also gain exposure to different caveats of the research and development (R&D) activities within the government research facility/laboratory. In addition, the intern will be directly mentored by leading phase-resolving numerical wave modeling experts in the field. The internship will culminate in a peer-reviewed journal, or a conference proceedings publication.

An ideal candidate should have a good knowledge of applied and computational mathematics (numerical modeling with both finite difference and finite volume schemes), with emphasis on nonlinear free-surface flows (water waves). It is also desirable, but not necessary, for the intern to be able to read and write code in lower (FOTRAN) and higher-level (Python) programming languages, as well as have a basic understanding of distributed parallel-computing paradigms (MPI – Message Passing Interface).

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

Hosting Site:

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

Internship location: Vicksburg, MS

Project Description:

Recent advancements in projection-based reduced-order models have enabled the construction of fast and approximate surrogate models for complex, parameterized nonlinear dynamical systems. In this project, we seek to augment these technologies by developing stochastic Petrov—Galerkin reduced-order models (SPG-ROMs) for uncertainty quantification. Like the stochastic Galerkin method, SPG-ROMs are promising as a single solve yields the entire posterior distribution for quantities of interest. In the SPG-ROM formulation, a parameterized dynamical system solution is represented by tensor products of data-driven basis functions in state-space, and polynomial chaos expansions in stochastic space. A reduced-order model is then obtained via orthogonality or residual minimization constraints. This project will focus on (1) development of the SPG-ROM formulation, (2) theoretical numerical analyses of the approach, and (3) the deployment of the approach to benchmark hyperbolic systems, on which classical approaches (e.g., Stochastic Galerkin) often fail to yield adequate results.

Preferred skills: Linear algebra, numerical methods, elementary statistics, proficiency in Python or C++.

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

Hosting Site:

Sandia National Laboratories

Internship location: Albuquerque, NM

U.S. Citizenship is a requirement for this internship

Project Description:

The discipline of hydro-climatology relies heavily on meaningful information retrieved from the data associated with it. Hydro-climatological data includes temporal (time series) and spatial (landscape) information. Spatio-temporal data helps in assessing the various levels of impacts of frequently occurring disastrous extreme events. Extreme events, including coastal flooding on the Southeastern Atlantic and Gulf Coastal Plain and forest fires on the west coast, have adversely affected economic resources and losses of enormous property and human lives. To improve the preparedness plan and reduce such losses, the protection and mitigation planning agencies want information before the natural disaster hits the ground. A mitigation plan can potentially be put in place with the velocity and volume of necessary information the hydro-climatological data is producing. This study aims to use the long-term climate data and information from USDA Forest Service Santee Experimental Forest (SEF) and assess the changes in climate over space and time at the study site. The observations would, then, be related to natural physical processes, including soil moisture and temperature and evapotranspiration for woodland fire risk assessment. The study's tasks include training the graduate student on Mathematical and Statistical modeling using R/Python; Data Mining, Analysis, Visualization, and Interpretation; Remote Sensing and Geographic Information System (RS & GIS); and Decision Making. The incumbent will be using the existing knowledge of computer programming, statistics, and engineering mathematics to enhance the skills to address the dynamic issues associated with the hydrological cycle and its components. The students will also learn about field experimental studies, forest hydrologic processes and tidal flow dynamics represented by mathematical equations, real-time monitoring technology, and managing and analyzing the Big Data sets using statistics at the host SEF study site.

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

Hosting Site:

USDA Forest Service, Center for Forest Watershed Research

Internship location: Asheville, NC

Project Description:

Climate extremes, such as hurricanes and tropical storm-induced rainfall, cause severe floods, which is expected to happen more frequently in the future due to climate change. Coastal communities and infrastructure systems, including water, energy, and communications, are particularly vulnerable to flooding caused by combined rainfall, sea level rise, and high tide. Communities are investigating and implementing flood risk mitigation and adaptation strategies, such as early warning systems and flood control structures, to reduce the flooding risk and damages. This research aims to enhance the response time and accuracy of flood detection, early warning and monitoring systems, by integrating the rapid growth in visual sensing technologies into flood data acquisition and modeling. This will be achieved by developing a visual sensing tool which measures the river depth and discharge using ground-based cameras and compare them with data from a watershed at USDA Forest Service Santee Experimental Forest (SEF). The objectives are a) collecting visual information from cameras and observed hydrologic data for the study site and b) extracting numerical information from semantic segmented images using a trained convolutional neural network (CNN). The tasks to achieve the goal are to i) develop program to extract on-site visual data to train CNNs, ii) conduct hydrologic analysis and develop stream network bathymetry, iii) to run trained CNNs and numerically integrate the information with the bathymetry to extract the water depth and computed discharge at different locations. Finally, the collected data will be used as inputs to a flood model developed for SEF to simulate and predict the streamflow discharge and flood properties. The students will learn about field experimental studies, hydrologic processes and flood flow dynamics represented by mathematical equations, real-time monitoring technology, and analyzing the Visual and Big Data using statistics at the SEF study site.

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

Hosting Site:

USDA Forest Service, Center for Forest Watershed Research

Internship location: Asheville, NC

Project Description:

Increased peak-flow magnitudes resulting from growing extreme precipitation events might have adverse effects on existing road drainage and culverts, resulting in their failures, increased flooding, soil erosion, economic losses, and disruption of stream connectivity critical for aquatic organisms. Engineers and hydrologists often use precipitation intensity-duration-frequency (PIDF) curves for design of such infrastructure. The goal of this study is to develop an online spatial hydrologic tool (Real-time Dashboards on Cloud) to generate a vulnerability assessment map for road culverts using long-term data from a low-gradient watershed at USDA Forest Service Santee Experimental Forest. The objectives are a) to identify erosion hazards and vulnerability risks to these structures using the PIDF-based design rainfall intensities and other geospatial data for the study site and b) to quantify the design discharge using widely used empirical methods. The proposed tasks are to i) develop a program to extract on-site long-term precipitation data to derive the PIDF distribution, ii) to conduct watershed hydrologic analysis using ArcHydro Tools to develop stream network and determine the potential culvert locations, iii) to use USDA-ARS RUSLE model and also develop a Stream Bank Erosion Vulnerability Analysis (SBEVA) model, both in ArcGIS ModelBuilder, to identify low, moderate, and high erosion vulnerable locations for determining the scale-based vulnerable culverts by combining results from both the models, and iv) link both the models in ArcGIS Pro (on Cloud) with design discharge estimating equations to assess capacities of culverts and develop a real-time Dashboard online culvert vulnerability assessment software. The students will also learn about field experimental studies, hydrologic processes and tidal flow dynamics represented by mathematical equations, real-time monitoring technology, and managing and analyzing the Big Data sets using statistics at the host site.

Disciplines: Analysis, Operations Research, Probability and Statistics, and Topology

Hosting Site:

USDA Forest Service, Center for Forest Watershed Research

Internship location: Asheville, NC

Project Description:

Projection-based reduced-order models are a class of physics-informed surrogate models suitable for large-scale nonlinear dynamical systems.  Traditionally, projection-based reduced-order models have involved projection of the system dynamics onto a low-dimensional linear space.  Unfortunately, for a certain class of physical systems, a suitable low-dimensional linear space does not exist.  For these systems, a low-dimensional nonlinear manifold is more appropriate.  In this work, we will examine various manifold learning techniques and their suitability for use with projection-based reduced-order models.  Of particular interest are algorithmic and architectural improvements for reducing offline and online computational costs associated with graph convolutional auto encoders.

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

Hosting Site:

Sandia National Laboratories

Internship location: Albuquerque, NM

Project Description:

Are you in interested in learning cutting-edge computational methods and algorithms for large-scale dynamical systems? This might be the right project for you!

This project focuses on advancing algorithms and computational methods for projection-based reduced order models (ROMs) of large-scale dynamical systems. State-of-the-art ROM formulations pivot around expressing the state as a rank-1 tensor (i.e. a vector) leading to computational kernels that are memory bandwidth bound and, therefore, ill-suited for scalable performance on modern many-core and hybrid computing nodes. We aim at exploring alternative formulations, e.g., rank-2, batched kernels and/or hierarchical parallel approaches, to overcome the memory bandwidth bottleneck and solve these problems efficiently. This would substantially impact the scalability and efficiency for solving many-query problems, e.g., those stemming from uncertainty quantification studies.
We expect the internship to include the following stages: first, we explore the current landscape of ROM formulations, and identify potential weaknesses and ideas for improvement; second, we identify one problem to target and lay out the mathematical and computational details to improve on it; third, leveraging a target test case (e.g., a simple PDE), we test these ideas in practice, documenting the benefits with an outlook to future directions.

Desired skills and experience: experience in applied mathematics and physical simulations, good knowledge of linear algebra, and basic knowledge of tensor algebra. Good knowledge of Python (possibly object-oriented) and basic knowledge of C++. Knowledge of GPU computing is a plus.

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

Hosting Site:

Sandia National Laboratories

Internship location: Albuquerque, NM

U.S. Citizenship is a requirement for this internship

Project Description:

The intern will support the “Reading The Ground” project through the development of classification algorithms and detailed performance comparisons. The activity will require some knowledge of multiclass classification techniques as well as some knowledge of the relevant performance metrics. The primary focus of the activity will be a detailed algorithm comparison and documentation. The intern will join a cooperative group and will participate in laboratory meetings and showcase results.

Disciplines: Mathematics (General), and Probability and Statistics

Hosting Site:

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

Internship location: Vicksburg, MS

U.S. Citizenship is a requirement for this internship

Project Description:

The intern will support the Modernizing Environmental Signature Physics for Target Detection project through the development of processes for the efficient analysis of large and disparate datasets.  The activity will require the integration of image files for EO and IR cameras. LIDAR data, soil sensor data, and meteorological data in order to perform data analysis. The intern will need a familiarity with multivariate and spatial data analysis and the development of MATLAB or R scripts to facilitate the data analysis or data retrieval in order to perform the analysis.  The project involves assessing the soil and meteorological conditions that hamper improvised buried object detection yielding inconsistent probability of detections and high false alarm rates. The intern will join a growing cooperative group and will assist many group members, participate in laboratory meetings, and showcase results.

The intern should have experience with algorithm development, coding, script development, machine learning, MATLAB, R, multivariate statistical analysis, and stochastic modeling as well as an interest in physics, mathematics.

Disciplines: Analysis, and Mathematics (General)

Hosting Site:

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

Internship location: Vicksburg, MS

Project Description:

This project studies corrosion inhibitor molecules in the form of coordination complexes with a data-driven approach that utilizes graph neural networks (GNNs) for a natural, intuitive featurization and embedding of molecular structures. Based on properties computed from high-fidelity atomistic simulation (e.g., density-functional theory), we will train molecular GNNs to predict properties of complexes that are relevant to corrosion inhibition (e.g., binding energy). Additionally, due to the combinatorically and virtually infinite space of ligand-metal coordination configurations, we will develop a combined molecular GNN with Gaussian process to form a “monolith” machine learning model that can be conveniently trained end-to-end. The uncertainty estimates from such model can help inform us to efficiently sample the space of the coordination configurations using sampling policies similar to that in Bayesian optimization techniques. Finally, this project has practical application in the area of high-throughput screening of molecules for ideal properties with respect to corrosion inhibition.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Livermore National Laboratory

Internship location: Livermore, CA

Project Description:

In this project, we will develop a novel data-driven, chance-constrained and risk-aware decision making under uncertainty framework.  The framework will be demonstrated on power system planning, operation, and control, for which we will strive to make the decision-making procedure applicable for different practical situations. This project will address the challenges related to the (1) computational efficiency of the algorithm, (2) the estimation accuracy in the uncertainty quantification, (3) the scalability of the proposed algorithm to a nonlinear system model, (4) the complexity in high-dimensional uncertainty modeling, and (5) the adaptivity in the decision-making. Through this studied decision-making under uncertainty procedure, the efficiency, security, economy and the risks of power system cascading failures can be properly managed to enhance the robustness and the risk-awareness in the operation. Specially, the tasks include (1) using traditional linear-transformation-based approach to explore the latent space for the surrogates, (2) using the proposed manifold-learning-based approach to explore the latent space for the surrogates, and (3) Comparing the performance of the manifold-learning-based approach with other dimension-reduction methods. To sum up, we will develop the surrogate-based decision-making under uncertainty framework in this project. The framework will not be limited to the applications in the power systems but providing great benefits in the control and design procedure of many engineering fields, such as the circuit design, air- craft design and control, robot control, vehicle and train control.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Livermore National Laboratory

Internship location: Livermore, CA

Project Description:

The goal of this project is to design and implement an optimization algorithm for model calibration. The algorithm seeks to identify model parameters whose values are not known and must be determined from indirect measurement via an inverse problem. The optimization problem to be solved uses a metric based on the earth mover's distance from optimal transport, which has been shown to be insensitive to noise in the data. Thus it may provide a better alternative to other approaches based on standard norms. Students will learn about regularization for inverse problems, tools from optimization, and topics from optimal transport.

Disciplines: Applied Mathematics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The goal of this project is to explore transport-based algorithms for improving the stability and training efficiency of neural networks. The project involves the approximation of neural networks as continuum equations, implicit discretizations of those equations, and iterative methods for solving them using tools borrowed from kinetic transport equations. Students will learn about neural networks, iterative methods for nonlinear systems, numerical tools for solving transport equations, and hyperbolic relaxation.

Disciplines: Applied Mathematics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

U.S. Citizenship is a requirement for this internship

Project Description:

Programmable logic controllers (PLCs) are used for industrial automation in many safety- and security-critical contexts. The IEC 61131-3 standard informally describes five languages intended for programming PLCs. Three of these are graphical -- ladder diagrams (LD), function block diagrams (FBD), and sequential function charts (SFC) -- while two are textual -- structured text (ST) and instruction lists (IL). And often, PLCs are programmed using a combination of these languages. As the scale and complexity of industrial automation grows, so do the scale and complexity of these programs. When these programs are deployed in a critical role, we would like mathematically-rigorous confidence in their correctness.

This project will explore techniques for high-assurance PLC programming in the IEC 61131-3 languages: formal verification of PLC programs or an IEC 61131-3 language implementation, model checking, abstract interpretation, or other static or dynamic analyses of IEC 61131-3 programs. Past MSGI projects have successfully published papers to top computer science and security conferences. Likewise, we plan to publish a paper on these findings in one of the high-ranking computer science (programming languages, formal methods) or cyber security conferences with the student as the lead author.

Disciplines: Applied Mathematics, Logic or Foundations of Mathematics, and Mathematics (General)

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

U.S. Citizenship is a requirement for this internship

Project Description:

Modern critical infrastructure (e.g. power grids, water and gas utilities, factories, etc.) face attacks by nation-states and advanced adversaries on a regular basis, yet few techniques exist to detect *cyber*-specific faults (i.e. problems) on the critical infrastructure devices. Given ORNL's existing platforms for building resilient cyber-physical systems and digital twin platforms for SCADA devices, this project seeks to classify relevant SCADA communications as corresponding to faults as opposed to normal operational behavior, like turning on or off power flow on purpose, rather than done by a malicious entity. The project will take advantage of modern data science, machine learning, and natural language processing techniques, or any technique of interest to the participant that could classify SCADA communications as corresponding to a true fault. 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 platform which will be deployed to utilities and live power grids in 2021. A successful student has prior experience with data science techniques and the basics of machine learning, but is not expected to have deep experience with programming. Notably, prior projects at ORNL by MSGI participants by this team have led to papers published at major computer science conferences. Based on the findings here, we will also seek to publish a paper in a major computer security venue with the participant as the lead author.

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

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The discovery of exoplanets (Nobel Prize 2019) and observations of proto-planetary disks (where exoplanets were born) have yielded rich information on the formation and evolution of planets and their systems. The primary building blocks of planets are gas and dust (Earth is made of mostly dust). As these components move around their proto-Sun, their dynamics can be modeled using advanced hydrodynamic simulations. But dust and gas move at different speed and their interactions can promote dust clumping, leading to condensation and eventual planet formation. These processes create "footprints" that are being observed by the most powerful telescopes in the world, giving us hints and hope to understand such complicated yet fundamental processes.

The objective is to help interns learn and use the state-of-the-art numerical methods and tools in modeling such multi-component systems. The interns will collaborate with scientists on exploring new numerical methods that can speed up the simulations of dust-gas evolution in protoplanetary disks. A fair amount of simulations on super-computers will be involved as well. The interns will also interact with other students, postdocs and scientists in a group environment that emphasize discussions, learning and collaboration. Regular activities such as hiking and cook-out are included as well.

Disciplines: Applied Mathematics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

This project will investigate the algorithmic scalability of neural network model training. Scalable training of neural network models allows researchers and practitioners to efficiently explore the effectiveness of larger models, larger data sets, and longer training epochs. Such a study is especially important for emerging model families in deep learning to understand their trade-offs between their algorithmic scalability and their expressiveness power. As an example, a graph neural network is an expressive and flexible neural network model family that can understand either the properties of the whole graph structure or the properties of individual constituents in graphs. Graph neural network models are actively investigated in DOE-mission relevant areas due to the state of the art accuracy in calculating energy levels in chemical, biological, and materials systems, with radically lower computational costs than Density Functional Theory-based approaches. However, similar to other emerging model families in deep learning, the trade-off between its algorithmic scalability and expressiveness power is still an open question. Including graph neural networks, such a gap creates challenges to use emerging neural network models in scientific areas.

Students will learn python-based deep learning platforms in high performance computing environments, with the empirical studies on public graph datasets. Student will also gain knowledge in discrete optimization, graph algorithms, and stochastic optimizations.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

We propose to develop methods that extract and construct reduced-order dynamics from the output of high-fidelity (HiFi) simulation of dynamical systems in a principled way. Given the dimensionality of the reduced configurational space, our aim to develop algorithms to identify a reduced-order dynamical model that optimally approximates the HiFi dynamics. We will leverage the mathematical structures developed in the Koopman von Neumann (KvN) formalism, in which fitting-to-data is always a linear regression and convex problem, but with the disadvantage that the operating space is an infinite-dimensional Hilbert space. We will address the problem of infinite-dimensionality by operating in the Mori–Zwanzig (MZ) projection operator formalism which provides mathematically accurate reduced-order dynamics and associated error, and we will use algorithms, including but not limited to machine-learning methods, to minimize the error. Initial applications will focus on reduced models of atomistic simulations of molecules and materials.

The intern will participate in a highly interdisciplinary research project which involves applied functional analysis (Koopman von Neumann and Mori–Zwanzig formalism), dynamical systems, optimization methods, and data-driven and machine-learning methods. The mentors will collaborate with the intern to formulate a sub-component of the project. Examples of sub-components include: (1) developing loss function(al) for optimizing the reduced-order model, (2) imposing the mathematical structure of KvN and MZ to off-the-shelf machine-learning architectures, such as recurrent neural networks, and (3) demonstrating the approaches on data produced by large-scale atomistic simulations.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Probabilistic modeling has unique advantages in solving complex problems. On the one hand, the sampling procedures in probabilistic computing, e.g., various Monte Carlo techniques, bypass the curse of dimensionality and deliver key statistical quantities of the high-dimensional systems of interests without solving the full system. On the other hand, a probabilistic reasoning and learning framework has the capability to allow noisy, uncertain, heterogeneous, or even sparsely collected data streams, and is favored in causal inference, uncertainty quantification, and data fusion.

As increasing amounts of heterogeneous data are collected, it is critical to develop data-driven methods to calibrate and construct large probabilistic models. Such an approach is currently challenging due to the absence of a scalable algorithm to extract the essential sensitivity information—the derivatives of the error measure with respect to each of the model parameters—which is required in gradient-based optimization and uncertainty quantification procedures for efficiently improving the performance of the models.

In this research project, we will explore two approaches to fill the void: (1) to develop importance sampling procedures for approximately solving the adjoint states which contain the sensitivity information, and (2) to leverage and generalize the "reparametrization trick" which is recently proposed in the field of probabilistic machine learning. We expect the developed algorithms will enable data-driven methods of parametrizing and constructing large-scale probabilistic models and will analogously play the pivotal role of the automatic differentiation which enabled deterministic deep learning practices.

The recruited intern will collaborate with the mentors to define a smaller-scale yet self-contained project which may lead to a successful scientific publication.

Disciplines: Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Fluids with complex equations of state (EoSs) have become increasingly important in energy systems. Examples of interest to NREL’s High Performance Algorithms and Complex Fluids (HPACF) Group include the use of supercritical carbon dioxide in high-efficiency energy systems that enable carbon separation, utilization, and storage (CCUS), liquid sodium for energy storage, biomass for energy applications, and low-temperature helium for energy-efficient cooling of quantum technologies. NREL is currently implementing these equations of state in a broad range of computational fluid dynamics (CFD) solvers, ranging from commercial codes to the high-fidelity open-source Pele combustion solver being developed as an application for exascale computing. The relative complexity of the equation of state used varies not only with the fluid, but with the application, and the temperature and pressure range of the system. This project will involve studying the impact of EoS choice on the stability, solution time, and physical accuracy of the solutions obtained from CFD solvers for realistic energy systems simulations. This is a relatively open field with significant opportunities for future activities and publication.

Disciplines: Applied Mathematics, and Mathematics (General)

Hosting Site:

National Renewable Energy Laboratory

Internship location: Golden, CO

Project Description:

The focus of this proposal is to consider two approaches for solving PDE problems with constraints: one based on Machine Learning (ML) using deep neural nets (DNNs) and the second based on a PDE constrained optimization approach using adjoint equations. From a mathematical perspective, a complicating feature of DNNs is
that there is little formal basis for analyzing exactly what the training of a DNN accomplishes. This question
of interpretability of DNNs refers to the question of what the selection of weights from training a DNN
means mathematically. On the other hand, the PDE constrained optimization approach is based on a direct space-time discretization of the underlying equations.

A recent approach to interpretability is to cast DNNs in the context of dynamical
systems or differential equations. The fundamental insight is that the propagation of information through
a DNN can be viewed as the discretization of a continuous (unknown) dynamical system much like a traditional PDE
solver. From this point of view, the design criteria of a DNN such as the number of layers, the size of the
data at each layer, and the activation function correspond to the number of time steps, the spatial resolution,
and the type of equation being solved. We would like to undertake a careful comparison of the two methods on a set of characteristic liner and non-linear PDEs in single and multiple dimensions to determine the efficiency and accuracy trade-offs of the two approaches.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

This project concerns the analysis and numerical evaluation of methods to extend spatial multigrid based methods for the implicit treatment of nonlinear diffusion terms in massively parallel PDE simulations.  Our approach combines highly-optimized second-order multigrid methods in the AMReX code framework with an iterative space-time deferred correction iteration to produce higher-order solvers designed for combustion applications.  The focus of this summer project will be investigating combining these solvers with the time-parallel library LibPFASST to expose further concurrency and reduce run times.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Modern deep learning models have proven to be highly effective at performing a wide range of discriminative tasks on complex data types, such as speaker detection in audio, object detection in images, and action detection in video. However, deep neural networks tend to be poorly calibrated (i.e. they do not estimate their own uncertainty well) [1], and they are vulnerable to a wide range of adversarial attacks [2]. These findings indicate that the methods are not effective at characterizing the true distribution of complex data types, such as natural images. Generative neural models, such as variational autoencoders and normalizing flows, show some promise in estimating complex data manifolds, but recent research has shown that they are ineffective at detecting out-of-distribution data [3].

This weakness in out-of-distribution detection is especially apparent when performing tasks such as misinformation detection, where we rely on subtle differences between synthetic, altered content and real content. In this project, we aim to characterize the difference in distribution of synthetically-generated images and audio from natural images and audio, and plan to develop robust methods for detecting fake data detection with unsupervised models. As an intern in this project, you will collaborate with a team of experts in machine learning and neuromorphic computing. The intern should have strong coding skills in Python, should be skilled in developing machine learning models, and should have a strong background in probability and statistics.

1. Guo, Chuan, et al. "On calibration of modern neural networks." (2017).
2. Carlini, Nicholas, Anish Athalye, Nicolas Papernot, Wieland Brendel, Jonas Rauber, Dimitris Tsipras, Ian Goodfellow, Aleksander Madry, and Alexey Kurakin. "On evaluating adversarial robustness." (2019).
3. Nalisnick, Eric, Akihiro Matsukawa, Yee Whye Teh, Dilan Gorur, and Balaji Lakshminarayanan. "Do deep generative models know what they don't know?.”

Disciplines: Probability and Statistics, and Topology

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

The Internet is notoriously vulnerable to attack by means of the Border Gateway Protocol (BGP). BGP attacks are escalating in frequency, severity, and sophistication, creating an urgent demand for the deployment of the next generation of real-time BGP anomaly detection technologies. This project aims to provide a basic understanding of how BGP works as well as its existing threats. The ultimately goal is to design and implement AI/ML based anomaly detection algorithms for better detecting incidents such as route leaking/hijacking and traffic interception attacks. The project includes theoretical and applied components that requires to write high level code to demonstrate its application is a practical setting. Learning objectives of the applicant include: (1) develop a basic understanding of time series base anomaly detection methods and its application to BGP anomaly detection; (2) acquire a set of analytical and computing skills for implementing the algorithms; (3) apply acquired skills for detecting BGP anomalies on real-time fashion.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The Electron Ion Collider (EIC) will be a future particle accelerator that will collide electrons with protons and nuclei to study the internal structure of particles. The electron beam will scan the arrangement of the quarks and gluons that make up the protons and neutrons of nuclei. The force that holds quarks together, carried by the gluons, is the strongest force in Nature. The EIC will allow us to study this "strong nuclear force" and the role of gluons in the matter within and all around us.[1]

The project encompasses the evaluation of beam backgrounds that may muddle up the physics signatures we are looking for.   It is an essential task to study the impact these backgrounds will have to the detector and any measurement at the EIC. Background studies will facilitate current experimental efforts to actively ensure that the machine design will not be adverse to physics in terms of background load. In this project we will focus on two major sources of background:  1. background due to protons in the beam interacting with residual gas in the beam pipe (beam-gas) and 2: photons arising from synchrotron radiation due to the electron beam. The internship will focus on synchrotron calculations, running and/or improving existing simulations and presenting the results in group[2] and collaboration meetings. The goal will be to interpret the results and provide an initial estimate of the background expected at regions about +-1m or more from the beam interaction region.  C++ or a similar computing expertise will facilitate the integration in the internship. An initial bibliographical task will be given at the beginning of the internship to familiarize the student with the physics of interest. Finally, our group has a number of experts committed to the project. The intern will work/tele-work with the mentor on a daily basis.

[1]https://www.bnl.gov/eic/
[2]https://www.lanl.gov/org/ddste/aldps/physics/subatomic-physics/henp-team.php

Disciplines: Analysis, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Deep Learning (DL) models are nowadays used in many scientific applications, e.g., for making predictions when mechanistic models are not available or too hard to use. However, blindly applying DL models to a dataset has its dangers, especially when critical decisions are based on predictions whose uncertainties are not properly quantified. Uncertainties in DL models can arise from a variety of sources and may influence what the best DL model type and architecture should be. In this project, your research will be to develop numerical optimization methods for identifying the best DL model architectures while taking into account the prediction uncertainties in order to obtain robust and reliable DL models. You will make use of a variety of tools including derivative free optimization and uncertainty quantification methods.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

In many scientific applications, computer simulations are used to approximate complex physical phenomena. These simulations often contain parameters that must be tuned in order to optimize an objective (e.g., maximize a performance measure, minimize an error metric). For these types of simulation optimization problems, we generally do not have an analytic description of the objective function or its derivatives (black-box) and evaluating the simulation is computationally expensive. In addition, if the simulation contains stochastic dynamics (evaluating the simulation for the same parameters gives different results), the simulation must be evaluated multiple times for the same parameters in order to obtain a statistically significant estimate of the response. Due to these challenges (computational expense, black-box functions, stochasticity), new derivative-free sampling algorithms that minimize the number of queries to the simulation must be developed in order to find optimal solutions efficiently.

In this project, your research will focus on the development of efficient and effective algorithms for solving these computationally expensive stochastic black-box optimization problems. This research requires the development of new sampling strategies that adaptively determine where in the parameter space the next simulation evaluation will be done and how many replicates of the simulation evaluation should be done. You will develop a suite of test problems to assess the performance of your developed algorithm and finally apply it to a real-world science problem.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

In many scientific applications, computer simulations are used to approximate complex physical phenomena. These simulations usually have parameters that must be adjusted in order to obtain the most accurate simulations. Accuracy is assessed by comparing the simulation output to observation data. However, these data are often noisy, and therefore parameter inference is needed to determine those simulation parameters that most likely explain the observations.

Simulations are often computationally expensive and may require several minutes or hours per run. Thus, during inference, we cannot query the simulation model thousands of times in order to find the desired parameter posterior distributions. Moreover, simulations are often provided as black boxes, i.e., there is no analytic description available and inference methods that are based on adaptive exploration of the sample space are needed. Previously, methods have been developed that exploit Gaussian process models as surrogates of the expensive simulation in Bayesian inference. However, these methods do not scale well with an increasing number of sample points and parameters.

In this project, your research will focus on the development of scalable inference algorithms that are efficient and effective for computationally expensive models. In order to achieve this, your will iproject nvolve the development of new sampling strategies that adaptively explore the potentially large-dimensional parameter space; the use of dimension reduction and sample space reduction methods; the use of Gaussian process models (or other types of surrogate models); and Bayesian inference methods. You will develop a suite of fast-to-compute test problems to assess the performance of your developed algorithm and finally apply it to a real-world science problem.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Solving flow and transport through complex geometries such as porous media involves an extreme computational cost. Simplifications such as pore networks, where the pores are represented by nodes and the pore throats by edges connecting pores, have been proposed. These models have the ability to preserve the connectivity of the medium. However, they have difficulties capturing preferential paths (high velocity) and stagnation zones (low velocity), as they do not consider the specific relations between nodes. Network theory approaches, where the complex network is conceptualized like a graph, can help to simplify and better understand fluid dynamics and transport in porous media. To address this issue, we propose a method based on eigenvector centrality. It has been corrected to overcome the centralization problem and modified to introduce a bias in the centrality distribution along a particular direction which allows considering the flow and transport anisotropy in porous media. The model predictions are compared with millifluidic transport experiments, showing that this technique is computationally efficient and has potential for predicting preferential paths and stagnation zones for flow and transport in porous media. Entropy computed from the eigenvector centrality probability distribution is proposed as an indicator of the “mixing capacity” of the system.

We propose to generalize this tool to three dimensions and produce a MATLAB based library for open source release. We are also envisioning interfacing LANL’s Basic Matrix Library (BML) with MATLAB for access to exascale HPC architectures. The Student will be exposed to the state of art of the HPC techniques, and cutting edge advances in microfluidics. Moreover, the student will be in contact with scientists from both LANL and ETH, two world leading scientific institutions.

Disciplines: Applied Mathematics

Hosting Site:

Los Alamos National Laboratory

Project Description:

Broadly speaking, the goal of the project is to use descriptors developed in Topological Data Analysis (persistence diagrams and merge trees) in machine learning. There are two possibilities: either one uses these descriptors as an input to machine learning algorithms or one uses these topological tools to regularize the ML models trained on a traditional input (e.g., a set of vectors from a Euclidean space of some fixed dimension).

A concrete direction that the project can take is regularization of neural networks using persistence-sensitive simplification. Here a student can investigate how different choices of topological simplification algorithms affect the performance of the regularized model.

Ideally, by the end of the project the student will have hands-on experience with topological descriptors and training of machine learning models. The project will involve a fair amount of coding and some experience in programming, in particular, in Python is desirable; knowledge of C++ will be beneficial, too, because the key topological algorithms are usually implemented in this language. Prior knowledge of (simplicial) homology and persistent homology will be helpful, but not required.

The project will probably begin with a reading phase, when a student will get some high-level understanding of all theoretical aspects (persistent homology, persistence diagrams, topological simplification, etc.). After that the student will implement different variants of simplification and perform experiments.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

There is a growing interest in large scale power system simulations that generate detailed, cycle accurate voltage and current signals within the transmission and distribution systems during large electro-mechanical transient events. These models are computationally intensive and could be significantly improved with new numerical methods targeted specifically at this problem. The proposed research would seek to discover new, efficient numerical methods for simulating alternating current circuits with sources that have time varying frequency and phase, and analysis of the generator and circuit interface model to identify modeling techniques that promote numerical stability. Experimental development of the new methods will be facilitated with benchmark models that are part of a new power system simulator being developed at Oak Ridge National Laboratory.

Disciplines: Applied Mathematics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

U.S. Citizenship is a requirement for this internship

Project Description:

The Kinematically Linked Model Framework (KLMF) uses the same mathematics found in 3D gaming (temporally linked quaternion based 6D transformations and ray tracing) to track multiple sensor components (e.g. bearings, encoders, read-heads, GPS, IMU, lenses, mirrors, and 2D/3D detectors) associated with optically based imaging systems (e.g. Lidar & Electrical Optical). KLMF provides the structure to design, develop, and link mechanical and optical models for rigorous propagation of optical field aberrations and lower level optical and mechanical misalignments to final 2D/3D products – revolutionizing the study of complex systematic biases and parameterization of calibration models in order to remove systematic 2D/3D image distortions. KLMF also enables scientists to visualize distortion propagation in 3D movies providing additional insights and study into the positioning of new sensors to automatically detect and remove systematic biases within the error propagation chain.

Students will learn how to study the calibration of imaging sensors using the following steps:

• HW Modeling: Use KLMF to model and link optical and mechanical components and propagate their errors to sensor device measurements: IMU, GPS, gimbal read-heads, 2D/3D detectors, etc.
• Sensor Modeling: Use KLMF to link sensor device measurements to create temporal meta-data estimates of 6D orientations of the 2D/3D detectors in world space.
• Sensor bias studies: Use KLMF to inject bias perturbations into HW component models to determine impact on systematic distortions within 2D/3D image products.
• Sensor calibration studies: Students learn how to create Sensor Model calibration parameterizations to remove 2D/3D image product distortions.

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

Hosting Site:

U.S. Army Corps of Engineers, Geospatial Research Laboratory

Internship location: Alexandria, VA

Project Description:

The intern would have an opportunity to collaborate with a large multidisciplinary team investigating wildland fire effects on plant tissue and subsequent whole plant mortality. The project includes both laboratory and field studies of energy dose dependent tissue damage to be able to both understand the mechanisms behind fire driven damage and mortality and be able to connect this knowledge to coupled fire-atmosphere models such as FireTEC and QUIC-Fire. Opportunities would include helping develop energy dose-damage relationships, novel ways of modeling 3D heat transfer, and fluid dynamics of heat flow through a tree canopy. Other opportunities would be available depending on the applicants areas of interest and expertise. Ideally the intern would participate at the USFS fire laboratory in Athens, Georgia.

Disciplines: Applied Mathematics, Biometrics and Biostatistics, and Probability and Statistics

Hosting Site:

USDA Forest Service Southern Research Station, Athens Forestry Laboratory

Internship location: Athens, GA

Project Description:

We seek to mitigate the effects of quantum errors in quantum simulation on gate-based universal quantum computers using machine learning. In particular, we want to mitigate errors in the time evolution of a neutrino scattering simulation (no knowledge of scattering physics is expected or required). Precise simulation of the quantum computer will soon be impractical, but even in the cases where we may simulate the device, building realistic noise models is extremely challenging. Therefore, we will study the applicability of machine learning as a heuristic algorithm, with the special advantage that we may train our error correction scheme on hardware directly and remove the need to build an accurate noise model.

Students will develop knowledge in using a quantum programming package (e.g. IBM’s Qiskit or Rigetti’s PyQuil) and develop a family of neural network algorithms (using either TensorFlow or PyTorch) to mitigate error in the simulation. Next we will study algorithm performance on real quantum computing devices.

This project will be conducted in a team setting under the primary direction of researchers at Fermilab, but with opportunities to interact with team members at the University of Washington. The entire project may be done remotely, with frequent video meetings and the use of other communication tools (e.g., Slack, email).

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Fermi National Laboratory

Internship location: Batavia, IL

U.S. Citizenship is a requirement for this internship

Project Description:

How do animal groups adapt their movement decisions to a changing environment of obstacles? Engineers have so far failed to create movement algorithms for autonomous groups that can successfully negotiate a potentially infinite set of environmental encounters. It is therefore extraordinary that nature has solved this problem with seemingly simple organisms whose behaviors are driven primarily by the need for food, to escape predation, or to reproduce. This presents a tantalizing opportunity to better understand the fundamental nature of communication between the individuals of a moving group that needs to sense, respond, and adapt to challenges and obstacles in their environment.

This internship will involve developing simple agent-based models of obstacle navigation, simulating those models computationally, and devising/testing various statistical metrics drawn principally from information theory to quantify inter-agent communication as obstacles are encountered and circumnavigated. These metrics will eventually be applied to trajectory data from experiments using both dermestid beetles and simple, non-interacting robots in order to ascertain the extent to which social interactions enhance navigational capabilities.

Disciplines: Applied Mathematics, Biometrics and Biostatistics, and Probability and Statistics

Hosting Site:

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

Internship location: Vicksburg, MS

Project Description:

Using models, COVID19 data as well as socio-economic models that represent the key drivers for social contacts, we will develop a probabilistic assessment of public health policies aimed at improving socio-economic resiliency of a  community faced with the prospects of an epidemic.

The novel aspect of this project is that we are going to apply ideas non-equilibrium statistical mechanics to formulate a scheme that evaluates health policies which are usually captured by models with many adjustable parameters. The product would be a metric that assesses the effectiveness (resilience) of a particular or a group of policies.

A second part of this project will look for ways to combine time dependent Bayesian estimation, the models and the data, along with learning networks, with the aim of formulating a linear response theory that can discover policy practices and characteristics (parameters) that lead to desired metric goals and report the cost of achieving these.

Disciplines: Applied Mathematics, Biometrics and Biostatistics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

Rapid and large amplitude ground deformation such as induced by large magnitude earthquakes are now routinely imaged by Interferometric Synthetic Aperture Radar (InSAR).  However, measuring smaller amplitude signals remains challenging due to atmospheric propagation delays which may exceed the signature of deformation in InSAR time series.

Although atmospheric correction methods improve our ability to observe slow and small (i.e. mm/yr) deformations, expert interpretation and a priori knowledge of deforming systems is always required to highlight deformation signals.

In our initial research, we developed a deep learning architecture tailored to remove atmospheric delays due to turbulence and layering of the atmosphere, as well as to identify and extract transient episodes of ground deformation.

In this project, you will be exploring probabilistic deep learning architectures and methods that can push InSAR data analysis towards reliable automation at a global scale.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Satellites have been observing the Earth for decades, with new satellites coming online at a rapid pace. These massive archives provide new opportunities to understand the distribution and dynamics of surface water across the planet, but this is only achievable through tools that can automatically and accurately extract waterbodies. Using a large training dataset collected over Arctic Rivers, we found a deep convolutional neural network (CNN) model showed great promise toward this goal, with higher accuracies and better boundary delineations than standard machine learning pixel-based classification approaches. We are aiming to build from this initial success by expanding the domain of the model (from Arctic to other regions), expand its application from Landsat-only to other satellite sensors, and improve boundary detection through super resolution techniques.

A candidate for this research would have interest applying deep machine learning to remotely sensed data. Our current codebase relies on the Python package Tensor flow, and we can provide training in using this package if you have no previous experience. Some Python programming experience is recommended. Specific project goals can be customized around your interest and, in addition to the problems mentioned above, can include applications of the CNN model to analyze surface water dynamics in Arctic regions (such as lake expansion and shrinkage, river migration, and/or delta channel morphodynamics) or more general problems such as reservoir monitoring. We would also like to build a model for high-resolution imagery such as Planet Labs that may contain only RGB bands.

Disciplines: Applied Mathematics, and Mathematics (General)

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Topological properties of physical systems that are those that are preserved under continuous deformations. Such properties which may manifest in vector fields and other tensor fields in Euclidean space responsible for many exotic properties of physical systems (e.g.  KT-transition). Preserving (i.e. encoding) the topology of such systems is important for building learnable models that generalize well across various physical systems.

In this project, we will construct modular neural network operations that preserve the topology of inputs and are able to transform inputs to outputs of different representations (e.g. vector field to a higher-rank tensor field) while preserving topological features. We will use test cases inspired by physical systems, e.g. spin waves on periodic lattices and vortices in 2D vector fields.

Recommended prerequisites (by start of internship): Knowledge of group theory and representation theory and how to articulate the tools of these theories in code. Familiarity with python, PyTorch, and the mathematical underpinnings of basic neural network operations (forward and backward propagation). Some familiarity with concepts in equivariant neural networks will be helpful  (see e3nn.org for an example).

Disciplines: Applied Mathematics, Geometry, Mathematics (General), and Topology

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Point geometry, especially the point geometry of atomic structures, can be hierarchical in nature and composed of geometric motifs that can reoccur in any location or orientation in a given example. It is an open question how to sample conditional probabilities of point-wise geometries in a way that preserves the underlying symmetry of 3D space.

In this project, we will construct methods for sampling point-wise geometries using Euclidean Neural Networks which are equivariant to elements of Euclidean symmetry (rotations, translations, and inversion). We will build upon the research in this reference: https://arxiv.org/abs/2007.02005 and determine the form of networks output needed for the sampling of degenerate outputs.

Recommended prerequisite (by start of internship): Familiarity with degeneracy as it manifests in quantum mechanical systems and concepts such as density matrices. Familiarity with python, PyTorch, and the mathematical underpinnings of basic neural network operations (forward and backward propagation). Some familiarity with concepts in equivariant neural networks will be helpful (see e3nn.org for an example).

Disciplines: Applied Mathematics, Geometry, and Mathematics (General)

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

The Metalog distribution (http://www.metalogdistributions.com/) is a flexible family of continuous probability distributions.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

In this project, we will explore 'spiking' algorithms for machine learning. These algorithms will be implemented on a neuromorphic computing system. Neuromorphic systems are designed to compute in a similar way to how the brain computes. That is, they are based on units that are meant to act like neurons in the brain and the neurons are connected in a massively parallel way. Implementing spiking algorithms is challenging and similar to the construction of a logical circuit. That is, neuromorphic computing is at the stage where we are identifying circuit mechanisms that we can use to make use of the parallelism available. These mechanisms consist of a library of useful concepts for performing important operations. Our Laboratory (Sornborger) has been working on such a 'programming' framework and we have a set of mechanisms that we use to implement algorithms. In this project, we will use these mechanisms (and possibly invent new mechanisms) to implement a spiking machine learning algorithm.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

The Computational Earth Science (EES-16) group at Los Alamos National Laboratory is looking for a highly motivated student with a strong background in applied mathematics and scientific programming to join us in the ongoing development and application of our discrete fracture network (DFN) software dfnWorks (2017 R&D 100 Winner – dfnworks.lanl.gov). DFN models sit at the junction of high-performance computing (HPC), numerical analysis, graph theory, and computational geometry. The student will collaborate directly with Los Alamos National Laboratory scientists to take part in the development of new capabilities in the software and using the new capabilities to probe critical questions at the frontier of our understanding of flow and transport through fractures. The project will include a variety of aspects of DFN modeling, such as machine learning, graph theory, computational geometry and meshing algorithms, as well as scientific applications of DFN models in the earth and environmental sciences. Specific objectives will be defined in collaboration with the selected student to fit their interests and background and there will be opportunities to collaborate and interact with other students, postdocs, and scientists in the group through weekly seminars and a research symposium. Last year, the successful candidate developed a new meshing algorithm for DFNs, that is now part of the production version of dfnWorks, and which is also forming the basis of a manuscript and part of their thesis research. For more information please contact Dr. Matthew Sweeney (msweeney2796@lanl.gov).

Disciplines: Applied Mathematics, Geometry, Mathematics (General), Probability and Statistics, and Topology

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Machine learning (ML) and artificial intelligence (AI) are transforming scientific research both within the Department of Energy and across academia at large. In particular, machine learning models based on tensor factorization have gained significant traction and seen widespread adoption thanks to their strong generalizability and interpretability. The proposed research concerns the acceleration of computing approximate tensor factorizations of high order data. To this end, we will investigate algorithmic primitives for tensor factorization that are scalable and friendly for massively parallel heterogeneous supercomputers, in particular those that are accelerated with general-purpose graphics processing units (GPUs).

Learning objectives & activities:

• Design and optimize parallel algorithms for tensor factorization.
• Implement GPU kernels using CUDA C++.
• Collect microbenchmarks and performance profiles.
• Draft technical reports for potential publication.

Disciplines: Applied Mathematics

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Recent advances in sparse linear algebra solvers suggest that the next step in scalable solver performance may be adaptive parameter estimation together with feedback and control approaches to optimize solver convergence rates. This approach would reduce the time to solution and overall model compute time. Because of the importance of sparse linear solvers to computational fluid dynamics (CFD) codes such as the Nalu-Wind code under development at NREL, these improvements would enable improved simulations of critical problems in renewable energy such as control of wind turbine wakes. Recent research in optimal control theory applied to the generalized minimal residual (GMRES) iterative Krylov solver shows these methods can lead to possible acceleration of linear and nonlinear solvers in CFD codes. Nonlinear solvers are typically embedded within a Picard iteration that can be accelerated with extrapolation schemes such as the Andersson algorithm. These solvers can also be accelerated using similar approaches and in combination with so-called Krylov recycling and optimal initial solution guesses. AI/ML can also be employed to learn the optimal solver parameters based on the flow physics of simulations. We propose to have a graduate student explore both aspects with a view towards improving the speed of HPACF science applications with a specific focus on wind energy technologies and the Nalu-Wind and AMR-Wind software packages.

Disciplines: Applied Mathematics, and Mathematics (General)

Hosting Site:

National Renewable Energy Laboratory

Internship location: Golden, CO

Project Description:

The GeoAI group at ORNL is hosting students in mathematics this summer to collaborate on a project that is developing novel private low-shot learning methods. Current areas of research include metric learning and metal learning for few shot; developing and incorporating privacy preserving methods into low-shot models to mitigate against membership inference and model inversion attacks; broadening the definition of membership inference attacks on various imagery domains; quantifying privacy loss due to membership inference or model inversion attacks; and understanding potential information loss in low-shot models. A participant on this project will, together with their mentor, have wide discretion in selecting a
problem of interest from a variety of topics in low-shot learning, private machine learning, mathematics and statistics.

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

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The GeoAI group at ORNL is hosting students in mathematics this summer to collaborate on a project that is developing novel private low-shot learning methods. Current areas of research include metric learning and metal learning for few shot; developing and incorporating privacy preserving methods into low-shot models to mitigate against membership inference and model inversion attacks; broadening the definition of membership inference attacks on various imagery domains; quantifying privacy loss due to membership inference or model inversion attacks; and understanding potential information loss in low-shot models. A participant on this project will, together with their mentor, have wide discretion in selecting a
problem of interest from a variety of topics in low-shot learning, private machine learning, mathematics and statistics.

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

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

Engineering and application-oriented mission science aims to alter system behavior to achieve specific objectives, or to make optimal decisions/predictions regarding system behavior. Many real-world applications involve highly complex systems which are computationally expensive to simulate and whose dynamics are substantially uncertain. Effective predictive science must often resort to surrogate models that represent a reduced form of the system dynamics, in order to explore the space of uncertainties in a more computationally tractable manner.

Machine learning (ML) has made data-driven models, such as deep neural networks, widely popular for learning such surrogates. However, complex systems are often highly nonlinear, while data for learning the surrogates are typically scarce and costly to acquire. Many “big data” models in the ML literature fall short of serving as adequate surrogates in this setting, and further fail to quantify scientific uncertainties in the system.

This project aims to develop Scientific ML techniques that enable objective-driven uncertainty quantification (UQ) for data-driven models. We will focus on developing theories and algorithms that can ultimately lead to an automated learning procedure of effective surrogates for complex systems that can be used for making optimal decisions robust to system uncertainties and surrogate approximation errors. These goals will be attained based on a Bayesian ML paradigm, in which we integrate scientific prior knowledge on the system and the available data to obtain a prior directly characterizing the scientific uncertainty in the physical system, quantify the uncertainty relative to the objective, develop optimal operators robust to the uncertainty, and design strategies that can optimally reduce the uncertainty and thereby directly contribute to the attainment of the objective. Potential applications of this methodology will be discussed with the student, but may focus on biological and biomedical discovery science.

Disciplines: Applied Mathematics, Biometrics and Biostatistics, and Probability and Statistics

Hosting Site:

Brookhaven National Laboratory

Internship location: Upton, NY

Project Description:

Deep neural networks have proven to be powerful tools for regression and function approximation, with well-known applications to image, video, speech, and natural language processing. More recently, researchers have begun to explore the application of deep learning methods to the solution of differential equations, with aims to produce more general-purpose numerical solvers or, potentially, to arrive at (possibly more approximate) solutions with greater speed. The applicability of these methods to geophysical fluid dynamics problems in weather and climate science is still poorly understood. This project will investigate the accuracy and efficiency of physics informed machine learning (PIML) algorithms on partial differential equation (PDE) test problems of interest to atmospheric and ocean dynamics, such as the quasi-geostrophic or shallow water equations. Of particular interest could be the recent mesh-free neural operator approaches that learn data-driven approximations to infinite-dimensional maps from PDE input spaces (boundary conditions or parameters) to output (solution) spaces directly, rather than by first passing through a fixed spatial discretization. The particular PIML methods to be used in the project will be determined in discussion with the student.

Disciplines: Applied Mathematics

Hosting Site:

Brookhaven National Laboratory

Internship location: Upton, NY

Project Description:

The concept of “measurement”, once restricted to one-dimensional features such as height, weight, density, voltage, lifetime, etc., has, over time, expanded to accommodate modern-day problems such as use of brain-scans to detect tumor, use of mass spectra and other types of spectra to identify drugs and other chemicals, use of fingerprints or footwear impressions to identify a pool of individuals who may have been present at the scene of a crime, use of DNA sequences to infer presence or absence of an abnormal gene, use of the chemical structure of a candidate therapeutic drug to infer efficacy in treating a patient, and so on. These modern problems have risen because of our ability to “sense” (or measure) more complex attributes of objects or entities than ever before. As a result, the fields of statistics, mathematics, and computer science are being challenged to keep pace and invent/discover more powerful approaches to measure, describe, and analyze complex, high-dimensional data.

Richard Royall (“Statistical Evidence: A Likelihood Paradigm”, 1997, Chapman & Hall/CRC) argues that, the degree of support provided by data in favor of a proposition A versus another proposition B is appropriately quantified by the ratio of the likelihoods for A and B, given the data. Accepting this point of view, interest turns to assessing likelihoods in any particular situation where evidential value is of interest. When the data are complex and high-dimensional (images, acoustic data, spectral data, shape data, etc.) assessment of likelihoods is extremely challenging partly due to the fact that modeling high-dimensional measurements requires a large amount of training data. In this project we will explore several currently available methods, including neural networks and deep learning, and transfer learning, for modeling high-dimensional data, especially in the context of data in the form of fingerprint images, footwear impression images, and spectral data.

Disciplines: Applied Mathematics, Biometrics and Biostatistics, and Probability and Statistics

Hosting Site:

National Institute of Standards and Technology

Internship location: Gaithersburg, MD

Project Description:

The Delaunay interpolant is a piecewise linear interpolant for finite datasets, which has been shown to have optimal accuracy in a sense with respect to other piecewise linear interpolants. Similarly, when taken to zero training error, a feed-forward neural network regressor with fully connected layers and ReLU activation functions is also a piecewise linear interpolant. The purpose of this project is to study the connection between Delaunay interpolation accuracy and neural network accuracy, potentially leveraging this connection to verify the accuracy of neural network regressors and detect inputs for which the neural networks may be subject to low-accuracy. Research activities may consist of mathematical analyses and experimental verification. This project is made feasible by the HPC resources available at Argonne and recent advances in Delaunay interpolation software, which allow for Delaunay interpolation in hundreds of input dimensions.

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

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

U.S. Citizenship is a requirement for this internship

Project Description:

Background: Lidar systems use the reflections of beams of light to determine the distance between the system and an object. One known issue with lidar systems is penetration through foliage. The human eye can often pass through foliage to discern what is on the other side, but even sparsely placed objects between the system and another object may interfere with lidar systems.

Objectives: The objective is to quantify the relationship between the density of “foliage-like” occlusion and the ability of a lidar system to discern what is beyond the “foliage.” This experiment will be used for the verification and validation of lidar system models in ERDC Modeling and Simulation (M&S) software.
Goals: The experiment must develop a consistent and quantifiable way to simulate foliage of various thicknesses and densities. Regard should be given to how foliage is modeled in ERDC M&S engines, so that these quantifications can be easily correlated to simulation environments. This experiment should yield sufficiently thoroughly quantified results that a realistic generation of LiDAR penetration through a variety of foliage can be modeled in ERDC’s M&S Engines.

Responsibilities and Learning Objectives: The student will fulfill all responsibilities for the completion of this project under the guidance of the researchers. The responsibilities of the students include but are not limited to: creating and improving experimental design, (reliably) implementing experimental set up, completing detailed and scientifically thorough testing and analysis, collecting and analyzing results, finding places to publish the research and conferences to present at, and writing publications. As such, learning objective include: design of experiments (DOE), statistical analysis, M&S/V&V exposure, publication and public speaking experience.  The researchers will provide funding, publication support (such as editing), coordination, and fill other supervisory roles.

Disciplines: Analysis, Applied Mathematics, Logic or Foundations of Mathematics, Operations Research, and Probability and Statistics

Hosting Site:

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

Internship location: Vicksburg, MS

Project Description:

This project seeks to develop a variational autoencoder inspired domain separation network framework to learn probabilistic multi-domain shared features. In many applications where intrinsic domain biases exist between source and target domain pairs, existing works have shown that the domain adversarial similarity loss is superior to extract shared features. However, when there are subtle differences between domains, as is the case in many remote sensing  and medical imaging applications, the extraction of shared features via the adversarial domain classifier may be prohibitive.

The goal of the project will be to develop a probability distribution inspired and distance-based similarity measure to learn a shared latent feature space. Variational autoencoders offer one way to establish a probabilistic latent variable model with rich information of the approximate posterior distribution. Our assumptions are that shared feature similarities can be computed between latent variables while taking into account the shape of the posterior distribution.

This probabilistic modification of existing domain separation networks approach can benefit from current work being performed in our related projects. The reserach includes variational methods for uncertainty quantification, as well as scaling the methods to prototype on the Summit supercomputing environment.  As an outcome we would like to enable the transfer of knowledge across object detection tasks to related domains for which shared embeddings must be compared per distributional features.

Disciplines: Applied Mathematics, Logic or Foundations of Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

Methods for multi-task learning that take advantage of natural groupings of related tasks are emerging across machine learning, computer vision and natural language processing communities. However the accompanying optimization techniques have so far demonstrated limitations when inter-task interactions are expected. In this project our interests are twofold: (1) designing new generic serial and parallel neural network architectures to enable multi-task models for Earth science challenges, (2) designing efficient optimization techniques to enable multi-domain learning with limited training data.

We define tasks groups to represent supervised information at the inter-task level that can be encoded into the model. For application we seek to efficiently learn different feature spaces at the levels of individual tasks, task groups, as well as the universe of all tasks while exploring the limitations of, (1) parallel architectures that encode each input simultaneously into feature spaces at different levels; and (2) serial architectures that encode each input successively into feature spaces at different levels in the task hierarchy. We have a globally distributed buildings dataset gathered from Terabytes of remote sensing images. The data is representative and diverse enough to explore hundreds of tasks in this project.

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

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The success of convolutional neural networks (CNN) and deep learning methods in solving a wide variety of difficult problems is well known. In classification or discrimination tasks involving 2D images the ability of CNNs to extract appropriate features from the training set of images has been well established. The realization that low level features extracted during the early layers of CNNs may be generalizable to image classes not part of the training set has led to ideas such as transfer learning. Notwithstanding such successes one might ask how efficient deep networks are in extracting the ‘correct set of features’ for a given task, say classification. The phrase ‘correct set of features’ can be given a definite meaning using ideas from classical statistics – ideas such as minimal sufficient statistics, complete sufficient statistics, etc. There are families of distributions for which a minimal set of features for discriminating between two or more distributions can be analytically derived. This minimal set is the absolute best data reduction possible without sacrificing discrimination power and is unique to within a one-to-one transformation. It is possible to produce realizations from these distributions in the form of images of arbitrary size (M x N pixels). Given two classes of images arising from two different parent distributions, some questions of considerable interest are: (1) What is the relationship between features extracted by a network and the theoretically optimal features? How does this relationship evolve as the number of training samples or network complexity increases? (2) To what extent does the discrimination efficiency (compared to theoretical optimum) depend on the type of architecture used for the network? (3) Can generative adversarial networks (GANs) be used in any way to gain efficiency? (4) Does classical statistics provide any insights regarding these questions? (5) Rather than pitting one classifier against another and comparing their performances, can we use challenging data sets where the best possible discrimination power is theoretically known, and any algorithm can assess its efficiency relative to this best case? A number of experiments will be conducted to answer these questions. Step 1: Identify classes of distributions that are capable of producing realizations in the form of images and for which the set of minimal sufficient statistics (minimal feature set) is analytically available. We have identified a few such classes for starters (multivariate Gaussian, multivariate Bernoulli, Ising Models) and we plan to consider other potentially interesting (but less well known) distribution families as well. Step 2: Identify a small number of candidate deep network architectures for use in the experiments. Step 3: Write software scripts for generating arbitrary (as many as needed) number of examples to form training sets (as many examples as the network needs for training it). Step 4: Train each candidate network using the training sets. Examine the features generated at various layers of the network, assess the degree of equivalence of the extracted features to the theoretically optimal feature set, and compute the discrimination power loss of the trained network pitted against the theoretical best feature set. Step 5: Study the ‘discrimination power loss’ as a function of number of training examples used and number of trainable parameters of the network. The proposed set of activities will provide answers to the questions posed at the outset and provide insights into the strengths, and more interestingly, limitations, of plug-and-play deep learning approaches. Testing deep learning approaches in situations where ground-truth optimal performance is known is an excellent way of understanding its performance characteristics. We view our efforts as the first step in creating certified image data collections with optimal features. This approach will shift the current culture of pitting one algorithm against another with respect to relative classification performance to a paradigm where each classification algorithm will be able to assess its efficiency by comparing itself to the theoretical limit of discrimination power.

Hosting Site:

National Institute of Standards and Technology

Internship location: Gaithersburg, MD

Project Description:

Time series signals are quite common in various physical systems. Particularly in this project, we will explore the acoustic time series collected from Acoustic Resonance Spectroscopy and Resonant Ultrasound Spectroscopy measurements.  Contained in both the spectral information and the time-series signals are signatures relating to material property, boundary condition, and/or dimensional/shape changes.  The mechanical vibrations from these experiments, and the signatures therein are used for nondestructive evaluation, material characterization and component sorting for inspection purposes of mission-relevant materials and components.

Effective extraction of the useful signatures and events from acoustic time series can be challenging in that they are limited in time duration, varied by signal amplitude, and corrupted by all sorts of environmental noise, as well as natural variations arising from the processing.   Simultaneously, it is necessary to limit the number of “false detections”  to a small fraction of the true detections.

The student will collaborate with both experimental acousticians and computational scientists to develop machine-learning algorithms to extract useful features and infer the events of interests. In order to gain the most out of our research project, we would expect students to have the following:     
• Proficiency in deep learning framework (PyTorch, TensorFlow, etc.)
• Expect to deliver scientific publications.
 
Reference
[1]. MC Remillieux, TJ Ulrich, C Payan, J Rivière, CR Lake, PY Le Bas, Resonant ultrasound spectroscopy for materials with high damping and samples of arbitrary geometry, Journal of Geophysical Research: Solid Earth 120 (7), 4898-4916

[2]. Yue Wu, Youzuo Lin, Zheng Zhou, David Chas Bolton, Ji Liu, Paul Johnson, "DeepDetect: A Cascaded Region-based Densely Connected Network for Seismic Event Detection," in IEEE Transactions on Geoscience and Remote Sensing, 57(1), 62-75, 2019.

Disciplines: Applied Mathematics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

Machine learning for scientific applications, ranging from physics and materials science to biology, has emerged as a promising alternative to more time-consuming experiments and simulations. The challenge with this approach is the selection of features that enable universal and interpretable system representations across multiple prediction tasks, and harnessing this understanding into designing new technologies.

In this project, we will develop tools from computational topology and geometry to study scientific systems, such as materials and molecular systems. We will combine these tools with machine learning to encode representations of these systems to aid in designing new systems to maximize desirable properties (such as those related to energy sustainability).

Recommended prerequisites (by start of internship): Knowledge of areas of topological data analysis and some of the code implementations related to this. Familiarity with python and PyTorch.

Disciplines: Applied Mathematics, Geometry, and Topology

Hosting Site:

Lawrence Berkeley National Laboratory

Internship location: Berkeley, CA

Project Description:

Some unique challenges in scientific data that need to be considered while building data-driven models are: (1) Noise and uncertainty (2) Data scarcity, in addition to the large feature spaces. Probabilistic models are a natural choice to address many of these challenges and provides a systematic approach to reason about the prediction uncertainty. Historically, the adoption of probabilistic modeling approaches has been limited by the scalability of the inference approaches. With the recent advances in Bayesian deep learning, especially in approximate inference approaches such as variational inference, we are able to efficiently handle flexible models (with millions of parameters) efficiently on modest hardware and obtain state-of-the-art predictive accuracy. The project would involve developing novel deep probabilistic machine learning approaches tailored to the unique needs of scientific data.

The participant will learn to develop deep generative models (that are at the intersection of Bayesian probabilistic modeling and deep learning) in the context unsupervised representation learning and supervised learning, for both Euclidean and non-Euclidean feature spaces as well as training them on leadership-class systems. Of particular interest would be to design efficient neural architectures and learning mechanisms through exploration of model hierarchy, topology preservation, contrastive learning, normalizing flows and careful neural architecture search for variational autoencoder and information bottleneck models. The participant will be part of a multidisciplinary team and will work on problems from different domains such as material science, high-energy physics and fusion energy sciences.

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

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

Wind conditions influence many human and natural systems (renewable energies, shipping, avia-tion, erosion, …) and extreme events of winds can have dramatic impacts on these systems. Wind speeds are typically measured as averages over a given time window. However, time-averaged wind speed misrepresents potential wind gusts (stronger winds usually recorded as a maximum over a time-window) happening during a given temporal window. Therefore, extreme wind events are not adequately captured by this data.

We propose to address the discrepancy between available measurements of time-averaged wind speed and wind gusts. We aim to develop hybrid techniques blending modern machine learning with classical statistics. The first facet of the project will consist of estimating and modeling features of the dependence between wind gusts and average wind speed with artificial neural networks (ANN) and deep learning (DL). ANN and DL have proven to reproduce complex nonlinear behaviors but have rarely been used in the context of extremes. This project connects the two emerging research areas, extreme event modeling and DL. In the second part, we propose to hybridize statistical models, that embed uncertainty associated with these phenomena, with predictions from ANNs.

Disciplines: Probability and Statistics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

We are interested in the development of optimal control models and algorithms for the simulation of quantum dynamics. We will investigate different sampling techniques and their impact on the performance of the optimal control techniques. We will also investigate the use of second-order information in the algorithms. This project requires familiarity with one or more of the following topics: optimal control, quantum computing, automatic differentiation, optimization methods.

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

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

We are interested in the development of optimization techniques that can make use of machine-learning (ML) surrogates trained on scientific applications within an outer loop design optimization and optimal control applications. This project will develop new optimization models and algorithms that enable design and control decisions over ML surrogates. Possible applications include the use of ML surrogates as regularization terms in inverse problems arising at Argonne's advanced photon source. Familiarity with one or more of the following areas is a plus: optimization methods, ML tools, and inverse problems.

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

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

Many interesting physical phenomena happen at scales finer than the discretization of numerical solvers can resolve; hence, the phenomena tend to be misrepresented in numerical solutions. We consider multi-scale problems based on partial differential equations (PDEs), where the presence of small localized features is challenging for numerical methods. At the same time, however, these fine-scale phenomena are critical to better represent the entire system. The numerical solution can be improved through several ways including: (i) adaptive mesh refinement and (ii) deterministic or stochastic subgrid-scale models. These approaches increase the computational or modeling efforts, respectively. Therefore, it is crucial to accurately identify the regions of the domain that require higher resolutions or adequate subgrid-scale models.

In this project, our goal is to develop new machine learning techniques, based on deep neural networks, to locate small-scale features of physical processes based on observable physical properties. This research will introduce the student to concepts from PDEs, statistics, numerical analysis, and deep learning, and we will explore ways of using techniques from these areas in creative and interdisciplinary ways.

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

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

Spiking neurons in the brain and spinal cord are typically modeled by systems of nonlinear ordinary differential equations (ODEs). For instance, one popular model, called Hodgkin-Huxley, is a nonlinear system of four ODEs. These ODE equations represent how spikes (electrical voltages) are generated in a neuron cell for a given current or sensor stimuli. The output of one such ODE takes the form of a spiking voltage time-series (similar to an electrocardiogram). These systems of ODEs contain numerous uncertain parameters that control the opening and closing of ion-channels on a cell membrane. Their exact values are not known in a prior, but rather their ranges are available from experiments. Therefore, the goal of this project is to numerically estimate the parameters' value such that the simulation output of the ODE fits spike recordings from laboratory experiments. This amounts to solving a so-called inverse problem. This can be done deterministically by solving an optimization problem; or in a statistical framework by estimating a posterior distribution for the parameters.

Computational challenges arise from the highly nonlinear and nonconvex loss function of the inverse problem, with sharp gradients and multiple local minima. Unfortunately, the loss cannot be regularized by a convex additive term since this would largely eliminate the information from data and model. Therefore, our goal is to develop new approaches to solve these challenging inverse problems in neuroscience. First, we aim to explore optimization algorithms that are capable of handling such nonlinearities and finding global minima, like Bayesian optimization. These methods are also used in the machine learning community. Second, we are interested in finding reconstruction operators based on training of deep artificial neural networks that approximate solution operators for our inverse problem.

The student gets the opportunity to work at the intersection of classical mathematics (systems of ODEs), neurosciences (brain and spinal cord), and modern machine and deep learning. The goal of the internship is to creatively explore machine and deep learning techniques to solve challenging inverse problems in neurosciences, while being part of a welcoming group of computational scientists at Argonne and neuroscientists from other universities.

Disciplines: Applied Mathematics, Biometrics and Biostatistics, Mathematics (General), and Probability and Statistics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

Highly flexible statistical models have the advantage of capturing intrinsic data characteristics such as spatial dependency or smoothness at various levels of extremes. However, some models are notoriously tricky to fit, even with relatively small data sets. On the other hand, machine learning techniques have proved successful in predicting complex systems or feature detection but are usually difficult to interpret and provide little uncertainty quantification. In this work, we propose to use machine learning techniques to estimate parameters and features of statistical models that are challenging to infer in the classical setups. The idea is that statistical models' interpretability still holds, and the machine learning tools become a stand-in for fitting purposes. Artificial neural networks will be trained using the statistical parameters or distribution features as output and the process's realizations as input data. Ideally, these parameters will have a distribution to characterize the uncertainty from the network output using a more statistical framework. Finally, we will explore the construction of distributions to characterize the uncertainty associated with the network outputs.

Disciplines: Probability and Statistics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

We are developing strategies for multivariate statistical modelling, in the context of model calibration and optimization, where the model response can be of arbitrary dimension or type. The project will entail (a) building surrogates for multi-output computer simulations, (b) discovering and modelling correlations among multi-output observational data, and (c) performing uncertainty quantification. Students will use multivariate modelling (parametric/non-parametric) apparatuses to achieve (a) and (b), and basic knowledge about model calibration is desired.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

Special functions may be defined as non-algebraic functions, including anything from simple trigonometric functions to Bessel functions. The accurate evaluation of such functions has always been problematic, the more common functions such as sines or exponentials may have compiler support, while functions like Bessel/Hankel require tailored libraries. Even so the evaluations may at times be inaccurate or inefficient.

Deep learning models have the ability to provide very efficient evaluations on graphical processing units, as well as provide more accurate values given the independence of the training set once the model has been developed. The end scope of this project is to develop the basis of a library for special functions in the language Julia, that outperforms traditional libraries and provides models portable to any other programming language. The work involves both theoretical analysis for determining optimal choices for model training, as well as programming for assuring the models are robust and portable.

Disciplines: Applied Mathematics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

Adjoint-based error estimation methods have been popular for estimating the global error in PDEs and ODEs. The basic idea is to use adjoint solutions to compute the inner product between a random vector and the global error at the final time. To estimate the pointwise global error at the final step, one would have to solve an adjoint system many times for each given vector. Existing methods allow us to evaluate the l2 error norm with a few random vectors. However, l2-norm is often not sufficient for studying the spatial distribution of error and providing systematic insights in error control. In this project, we will take a novel approach to recover the global error. We will identify the problem as a compressive sensing problem and exploit efficient strategies to reduce the number of random vectors needed. An alternative direction is to develop adjoint-based error estimation in multirate time integration methods. The participant does not need to develop adjoint models from scratch, the research will make use of the adjoint ODE solvers available in PETSc.

Disciplines: Applied Mathematics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

Sparse linear systems are a foundational component in modeling and simulation. As the workhorse of these applications, solution of sparse linear systems has been continuously driving the development of efficient iterative algorithms and their high-performance implementation. Because of the overwhelming choices of algorithms, data structures, and hardware architectures, selecting the optimal solution to a given system is difficult for application developers and researchers and may require deep skills and expertise in numerical analysis, HPC, and domain knowledge. In this project, we aim to automate the process and ease the reliance on numerical analysis and hardware knowledge by using graph neural networks (GNNs) for fast selection and tuning of preconditioners and sparse linear solvers adaptive to matrix properties and hardware architectures.

Disciplines: Applied Mathematics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

In many scientific domains, the data stems from different sources. Fusing the data from these sources is critical for many machine learning tasks. In this project, we will focus on data fusion methods for spatial temporal graph neural networks. Specifically, we will develop representation learning methods using encoder-decoder neural networks to model and learn the joint distribution of the modalities. We will use the trained generative model to sample data from the modality where the data is limited. We will evaluate the efficacy of the developed method on large scale traffic forecasting problems.

Disciplines: Analysis

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

The quantum Fourier transform (QFT) promises an exponential speed-up over its classical counterpart. In this project, the QFT will be assessed for utilization in pseudospectral methods for numerically solving partial differential equations (PDE). Currently, pseudospectral methods require several forward and inverse classical Fourier transforms for the computation of the nonlinear terms in a PDE. In this project, the QFT will be used to assess potential improvements in the scaling of these techniques. This will be through an analysis of the computational cost of the QFT in a PDE solver setting, as well as studies of practical deployments on quantum hardware.

Disciplines: Applied Mathematics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

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

Internship location: Lemont, IL

Project Description:

Developing a new optimization solver for a particular class of optimization problems is an expert-driven, iterative, and time-consuming process. While the process requires the knowledge and experience of the optimization expert, a major bottleneck stems from the trial-and-error approaches involved in trying out different initial ideas and algorithmic components, computationally expensive hyperparameter tuning, and extensive testing and validation across problem instances. In this project, we will develop learning to optimize approach, an emerging approach that adopts machine learning methods to help automate the design and development of optimization solvers and methods. Specifically, we will focus on designing mixed precision stochastic solvers using reinforcement learning methods and evaluate its efficacy to optimize neural network training.

Disciplines: Applied Mathematics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

The project will focus on developing a universal mathematical and computational framework for handling large-scale hyperbolic net and network problems. The basic computational methodology will be the discontinuous Galerkin method, with novel design of numerical fluxes in handling coupling conditions at network junctions that preserve mass conservation/balance of forces, continuity of solutions, as well as physical invariances such as positivity and well-balance properties of numerical solutions. In addition, the project will explore the development Eulerian-Lagrangian solvers for 1D and 2D shallow water systems with positivity preserving and well-balance preserving properties. Such solvers will be potentially integrated into the hyperbolic net/network to handle junction conditions going beyond the simple 1D algebraic coupling of solutions at different branches. The project will also address the challenge of the multiscale nature of the modelling levels and computational mesh levels.

The mathematical development will be implemented in the high-performance PETSc/DMNetwork libraries (https://www.mcs.anl.gov/petsc) for applications such as shallow water canal flows, blood flows, and the deformation of elastic nets for medical devices such as stents. The outcomes of this project will benefit the scientific community at large.

Disciplines: Mathematics (General)

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

Mixed-integer optimal control problems are a mathematical modeling tool with many applications ranging from the design of electromagnetics cloaks over energy management of buildings to image reconstruction. Solving mixed-integer optimal control problems on the other hand is a difficult task because the problems often combine both the challenges from PDE-constrained optimization and discrete optimization. Therefore, efficient approximation techniques have gained interest in recent years and we have contributed to this research area by developing approximation algorithms that guarantee certain degrees of performance.

In the proposed project, the student will be working on novel approximation algorithms for mixed-integer optimal control problems. The tasks include the development, analysis, implementation and testing of the algorithms on academic and real-world problem instances.

Disciplines: Applied Mathematics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

Simulations of complex physical phenomena is prevalent in many applications important to the Department of Energy, including climate sciences, high energy physics, and combustion research. These simulations usually contain parameters that determine how well the simulation represents reality. Optimizing these parameters is a computationally expensive task as it may take several minutes to hours to run the simulations with a given parameter set. Efficient optimization algorithms that do not rely on derivative information of the simulation objective function are needed. Surrogate model algorithms are commonly used to tackle these types of black-box expensive optimization problems.

To obtain the surrogate models, analysis is required to extract features such as the parameter dimensions that are variant to the simulations and to determine the degrees of freedom in the surrogate model so that it can fit the simulation data with high accuracy. Additionally, different physical aspects of the simulation may live in parameter subspaces of different dimensions. Hence, this project will focus on the use machine learning and statistical techniques to determine problem and simulation features that can be exploited in the optimization problem that fits the surrogate model to simulation data.

Disciplines: Analysis, Operations Research, and Probability and Statistics

Hosting Site:

Argonne National Laboratory

Internship location: Lemont, IL

Project Description:

This project is focused on modeling and simulation of nonlocal equations in the context of multiscale/mechanics problems. One of the most important open problems in this field is the identification of ``kernel functions’’ characterizing nonlocal operators. This non-trivial and ill-posed problem raises many mathematical and computational challenges. We plan to tackle model identification by combining 1) powerful tools of machine learning and physical principles and 2) versatile surrogates (e.g. neural networks) and a generalized nonlocal vector calculus that provides a universal definition of nonlocal operators. More specifically, we plan to develop a data-driven generalized tool based on approximations of the kernel function with neural networks, radial basis functions, or other complex surrogates. The outcome is a unified framework for new-model discovery.

Disciplines: Applied Mathematics, and Mathematics (General)

Hosting Site:

Sandia National Laboratories

Internship location: Albuquerque, NM

U.S. Citizenship is a requirement for this internship

Project Description:

Accurate measurement of shock wave fields with traditional pressure sensors is impossible under certain conditions, due to sub-microsecond shock rise times. Wave fields with known radial symmmetry can be reconstructed with higher accuracy from the optical phase difference induced in a Mach-Zehnder interferometer probe beam through an Abel transform relation. However, when the acoustic field departs from constrained spherical or cylindrical symmetry, additional unknown geometrical and physical parameters are introduced which confound the inversion. To overcome this limitation, we propose to develop approximate methods for local pressure field inversion from multiple, spatially-displaced interferometer probe beams. Physical and mathematical properties of the acoustic pressure perturbation can be used to develop displacement-dependent relations for optical phase. Based on these relations, the student intern will develop analytical and numerical methods for pressure inversion by exploiting the acoustic field similarity. In addition, the student intern will examine the sensitivity of the reconstruction to the diameter, profile, and positions of the interferometer probe beams.

The student intern will collaborate with a team of researchers at the U.S. Army Engineer Research and Development Center on this project. By the conclusion of the project, the student intern will be familiar with the principles and applications of the Abel and related integral transforms, analytical properties of acoustic wave fields, and principles of Mach-Zehnder interferometry. The student intern should have introductory coursework in numerical methods and Fourier analysis, and be proficient in a higher-level programming language, such as Python, for implementing numerical methods.

Disciplines: Applied Mathematics

Hosting Site:

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

Internship location: Vicksburg, MS

Project Description:

The goal of this project is to design and implement a deep neural network framework to infer the dynamics of physical systems from data, which also identifies and respects fundamental properties such as conservation and invariance. We will explore a particular class of neural networks, known as Hamiltonian neural networks (HNN), which discover conserved quantities via parameterizing the Hamiltonian of the system with neural networks and then learning it directly from data. We will examine properties of Hamilton’s equations as well as the class of symplectic numerical schemes to adapt and inform HNN designs. In this project, the student will learn about neural networks, Hamilton mechanics and how to apply them to extract the physical laws from scientific data, in particular, from fluid simulations and particle systems. 

Disciplines: Analysis, and Applied Mathematics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

Motivated by an ongoing project to help NIST engineers to design a cooktop fire warning system based on some pre-selected sensors to collect cooktop gas signals that may help predict potential fire based on a number of experiments that have already been completed, in this project student will explore the general issue of classification in the context of time-varying time series and how dynamic classification may be achieved using standard statistical discriminant analysis techniques such as logistic regression or recursive tree-based methods (using R) based on some multivariate features. Performance of these results may be compared with other machine learning methods such as neural networks that have already been tried but are known to have their own limitations such as unrealistic experimental data size requirements.

Disciplines: Probability and Statistics

Hosting Site:

National Institute of Standards and Technology

Internship location: Gaithersburg, MD

Project Description:

In recent work, we used compartmental epidemiological modeling and Markov chain Monte Carlo (MCMC) sampling to make daily Bayesian inferences to improve situational awareness of COVID-19 transmission in the 15 largest US metropolitan areas [Lin YT, Neumann J, Miller EF, Posner RG, Mallela A, Safta C, Ray J, Thakur G, Chinthavali S, Hlavacek WS (accepted) Daily forecasting of new cases for regional epidemics of Coronavirus Disease 2019 with Bayesian Uncertainty Quantification. Emerging Infectious Diseases]. Prediction of future new case detection with rigorous quantification of uncertainty was extremely computationally challenging, which has motivated an interest in developing software tools that facilitate efficient and scalable Bayesian inference that would be useful in future modeling studies. The student working on this project will become engaged in design, software implementation, and evaluation of MCMC algorithms. The goal is to implement one or more, potentially novel algorithms that are as efficient as possible and also scalable, meaning that computational cost does not grow explosively as the number of input/training data points or number of adjustable model parameters grows. The new methods will be incorporated into PyBioNetFit [Mitra ED, Suderman R, Colvin J, Ionkov A, Hu A, Sauro HM, Posner RG, Hlavacek WS (2019) PyBioNetFit and the Biological Property Specification Language. iScience 19: 1012-1036], which is an open-source Python software package designed to aid modelers in leveraging parallel computing resources to solve hard statistical inference problems. It is hoped that the student joining this project will apply the tools that they help to develop in a modeling study of a complex system of interest to them. The Hlavacek research team develops software for modeling of cell signaling systems and collaborates with quantitative experimentalists to study cell signaling systems involved in cancer and immunity. Our research is supported by grants from NIGMS and NCI. The student working on this project will learn about ODE modeling of nonlinear dynamics, parallel computing, Bayesian inference, and uncertainty quantification.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Los Alamos National Laboratory

Internship location: Los Alamos, NM

Project Description:

The internship will entail activities with the Athens Fire Lab of the Southern Research Station of the USDA Forest Service, in Athens, GA. The intern would gain experience collaborating with several Forest Service scientists, graduate students, and interns in Wildland Fire Science. This research is critical as wildfires are expanding and prescribed fires are becoming more important for mitigating wildfires and maintaining ecosystem health. An important part to understanding fire, is understanding the role of fuel or in this case vegetation, for driving fire behavior. We hope to utilize the intern’s mathematical expertise to advance our analysis, predictions, or modeling approaches for characterizing 3D forest vegetation structure and how it relates to physical properties of wildland fire, such as heat transfer and interactions with wind and fuel moisture properties. We aim to understand the mechanistic links between vegetation and fire to more accurately predict fire effects and feedbacks with fire-atmosphere dynamics. Mathematical relationships between multi-dimensional information, i.e. spatial and temporal changes in fire spread and vegetation (before, during and after fire) are also of interest. Ultimately, we will collaborate with the intern’s skills and interests to design an achievable goal for the internship within the Lab’s scope of work. The intern would likely utilize remote sensing data, such as 3D laser scanning (LiDAR: Light Detection And Ranging), infrared thermography and hyperspectral imagery, and use their associated instruments in a laboratory or field setting. There will be opportunities to visit forested field sites in the southeast to learn about wildland fire management, forest and fire ecology, prescribed burning practices and experimentation, or deployment of remote sensing instrumentation. During this internship, all safety standards are set high and COVID restrictions may limit in-person interactions and field experience.

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

Hosting Site:

USDA Forest Service Southern Research Station, Athens Forestry Laboratory

Internship location: Athens, GA

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

Internship location: Los Alamos, NM

Project Description:

The number of intelligent systems around us is growing rapidly. These Internet of Things (IoT) devices include smart home devices, health monitors, autonomous vehicles, and the smart grid, collecting data about our home activities, our health, where we visit, and our electricity usage, respectively. These technical means are constantly growing in power and sophistication and will likely see even more rapid development with the widespread deployment of 5G wireless networks, which will provide high speed data transfer and more precise location information. However, as these systems scale up, privacy is being left behind. We currently lack the ability to ensure meaningful data privacy guarantees to citizens, institutions, and infrastructure. And, we ask the question of how data privacy should be protected in a world where data is gathered and shared with increasing speed and ingenuity? Differential privacy (DP) is a new model of cybersecurity that proponents claim can protect sensitive data far better than traditional methods. Until recently differential privacy had been a topic of theoretical research without much application to real-world scenarios. So, there is a huge gap between theoretical bounds and practical implementation which opens the possibility for experiments. The aim is to create mathematically provable guarantee of data privacy protection and validate on real-world dataset related to smart grid to address the potential privacy consequences in those systems.

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

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The topic of this project is the development and study of events in dynamic systems that are able to deal with causal reasoning. Learning systems need to behave desirably in always changing environment settings, so they must gain causal understanding of their environment. This project has two aims (1) to use causal inference to model causality to help understand better causes, impacts and relationships so the system can make better decisions and (2) to advance the underexplored intersection of machine learning and causality. We will apply the methods in real-world data and evaluation.

Disciplines: Analysis, Applied Mathematics, and Probability and Statistics

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

As scientific experiments and High-Performance Computing (HPC) infrastructure evolve, data capture rates continue to exceed the available storage, network, and compute infrastructure for subsequent post-processing. The scientific data challenge is similar but distinct to many of the “Big Data” challenges we see in the commercial space. The trend from the newest diagnostics and exascale computations clearly shows that advanced machine learning techniques are necessary to manage, reduce, refactor, and extract information.

The project will focus on applying various machine learning and deep learning techniques for analyzing scientific data. The main research goals are i) how to analyze scientific data and apply machine learning algorithms for performance improvement, ii) researching advanced machine learning techniques for faster and more accurate models, and iii) how to automate science machine learning and deep learning workflows.

The project will provide the following learning opportunities; i) develop a basic understanding of scientific data processing workflows, ii) acquire skills in applying machine learning algorithms, and iii) gain experience in managing large-scale scientific data.

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

Hosting Site:

Oak Ridge National Laboratory

Internship location: Oak Ridge, TN

Project Description:

The overall objective of this framework is to enable complex interactions between physics-based and data-driven systems to be accounted for in real-time decision making while retaining credibility in mitigating rare events. This project will develop a data-driven surrogate model assisted deep reinforcement learning (DRL) framework to achieve fast and uncertainty aware decision makings. It has the following tasks: Task 1: Data-driven surrogate modeling and chance constraints reformulation: We will develop sparse Gaussian process (SGP)-based surrogate model to describe the relationships between uncertainty resources and the chance constraints. In particular, the SGP surrogate will be decomposed into two stages: the statistical-moment-based, i.e., the mean and standard deviation, rough approximation, and the error processing strategy to reduce the learning complexity. This allows us to achieve an accurate reformulation while retaining useful statistical moments information. Note that no priori distribution assumption is needed for uncertain variables. Task 2: Integrated SGP surrogate model and DRL algorithm for fast decision making: to enable a good performance, nonlinear SGP is usually required, yielding non-convex chance constrained OPF. This significantly increases the difficulty for nonlinear programming methods in getting good solutions and achieving fast decision makings. We will develop new safe DRL algorithms, i.e., safe actor-critic network that can continuously interact with SGP surrogate model and train an agent to learn the optimal control strategies. The direct interaction with the surrogate model instead of the original complex physical model significantly improves the training speed. Once the training is done, the agent is able to make fast control decision with new input variables, i.e., forecasted DERs and loads.

Disciplines: Applied Mathematics, and Probability and Statistics

Hosting Site:

Lawrence Livermore National Laboratory

Internship location: Livermore, CA

Project Description:

Collocation to solve the electronic Schreodinger equation is very attractive but has not been widely used in practice. The method replies on having a good set of trial functions to expand the unknown solution but does not require the calculations of matrix elements which can be very difficult for many basis sets. If one samples the solution on a 3D grid, the result is a matrix equation which can be rectangular in structure. Approaches such as the QR decomposition, the SVD and/or least squares can be applied to get the unknown coefficients in the trial function. The choice of grids can play an important role in a practical implementation of this approach and for molecules the design of an appropriate grid is not totally obvious. The student would be required to develop and perform numerical experiments that would look at appropriate basis sets and grids for some relatively simple diatomic molecules to ascertain whether the collocation approach can be made practical to compute the energy levels of these molecules. This could have important consequences for more complex systems where the difficulty lies in the evaluation of the Hamiltonian matrix elements using complex trial functions.

Disciplines: Applied Mathematics

Hosting Site:

National Institute of Standards and Technology

Internship location: Gaithersburg, MD

U.S. Citizenship is a requirement for this internship

Project Description:

The construction of a three dimensional grid which respects the local and global symmetry of a polyatomic molecule is not a trivial task. Near each atom, there is approximate spherical symmetry. Outside the bonding region, things begin to look spherically symmetric in a coordinate system whose origin is near the center of charge. At intermediate distances one gets contributions from the atoms and the central grid. There are approaches which divide space into "fuzzy" cells which use these different grids weighted by some function which divides the mesh into separate meshes for the atoms and the central grid and then carries out integration over each subregion summing at the end to get the final integral. The construction of the weighting function is a critical aspect to achieve efficiency. In this project, we would like to examine various weighting functions to find one that results in minimum number of points to achive chemical accuracy. Some work along these lines has been done by Axel Becke and others but they did not deal with cses where the electrons can escape the atoms as in scattering problems. It is these problems in which we have the most interest.

Disciplines: Applied Mathematics

Hosting Site:

National Institute of Standards and Technology

Internship location: Gaithersburg, MD