Publications

We present a method for learning dynamics of complex physical processes described by time-dependent nonlinear partial differential equations (PDEs). Our particular interest lies in extrapolating solutions in time beyond the range of temporal domain used in training. Our choice for a baseline method is physics-informed neural network (PINN) because the method parameterizes not only the solutions, but also the equations that describe the dynamics of physical processes. We demonstrate that PINN performs poorly on extrapolation tasks in many benchmark problems. To address this, we propose a novel method for better training PINN and demonstrate that our newly enhanced PINNs can accurately extrapolate solutions in time. Our method shows up to 72% smaller errors than existing methods in terms of the standard L2-norm metric.

Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The numerical scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for a linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme, which is stable. In addition, assembling the stiffness matrix of the nonlocal problem requires costly computational resources because high-order Gaussian quadrature is necessary to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares.

Second-order optimizers hold intriguing potential for deep learning, but suffer from increased cost and sensitivity to the non-convexity of the loss surface as compared to gradient-based approaches. We introduce a coordinate descent method to train deep neural networks for classification tasks that exploits global convexity of the cross-entropy loss in the weights of the linear layer. Our hybrid Newton/Gradient Descent (NGD) method is consistent with the interpretation of hidden layers as providing an adaptive basis and the linear layer as providing an optimal fit of the basis to data. By alternating between a second-order method to find globally optimal parameters for the linear layer and gradient descent to train the hidden layers, we ensure an optimal fit of the adaptive basis to data throughout training. The size of the Hessian in the second-order step scales only with the number weights in the linear layer and not the depth and width of the hidden layers; furthermore, the approach is applicable to arbitrary hidden layer architecture. Previous work applying this adaptive basis perspective to regression problems demonstrated significant improvements in accuracy at reduced training cost, and this work can be viewed as an extension of this approach to classification problems. We first prove that the resulting Hessian matrix is symmetric semi-definite, and that the Newton step realizes a global minimizer. By studying classification of manufactured two-dimensional point cloud data, we demonstrate both an improvement in validation error and a striking qualitative difference in the basis functions encoded in the hidden layer when trained using NGD. Application to image classification benchmarks for both dense and convolutional architectures reveals improved training accuracy, suggesting gains of second-order methods over gradient descent. A Tensorflow implementation of the algorithm is available at github.com/rgp62/.

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Poisson Tensor Factorization (PTF) is an important data analysis method for analyzing patterns and relationships in multiway count data. In this work, we consider several algorithms for computing a low-rank PTF of tensors with sparse count data values via maximum likelihood estimation. Such an approach reduces to solving a nonlinear, non-convex optimization problem, which can leverage considerable parallel computation due to the structure of the problem. However, since the maximum likelihood estimator corresponds to the global minimizer of this optimization problem, it is important to consider how effective methods are at both leveraging this inherent parallelism as well as computing a good approximation to the global minimizer. In this work we present comparisons of multiple methods for PTF that illustrate the tradeoffs in computational efficiency and accurately computing the maximum likelihood estimator. We present results using synthetic and real-world data tensors to demonstrate some of the challenges when choosing a method for a given tensor.

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This report describes the high-level accomplishments from the Plasma Science and Engineering Grand Challenge LDRD at Sandia National Laboratories. The Laboratory has a need to demonstrate predictive capabilities to model plasma phenomena in order to rapidly accelerate engineering development in several mission areas. The purpose of this Grand Challenge LDRD was to advance the fundamental models, methods, and algorithms along with supporting electrode science foundation to enable a revolutionary shift towards predictive plasma engineering design principles. This project integrated the SNL knowledge base in computer science, plasma physics, materials science, applied mathematics, and relevant application engineering to establish new cross-laboratory collaborations on these topics. As an initial exemplar, this project focused efforts on improving multi-scale modeling capabilities that are utilized to predict the electrical power delivery on large-scale pulsed power accelerators. Specifically, this LDRD was structured into three primary research thrusts that, when integrated, enable complex simulations of these devices: (1) the exploration of multi-scale models describing the desorption of contaminants from pulsed power electrodes, (2) the development of improved algorithms and code technologies to treat the multi-physics phenomena required to predict device performance, and (3) the creation of a rigorous verification and validation infrastructure to evaluate the codes and models across a range of challenge problems. These components were integrated into initial demonstrations of the largest simulations of multi-level vacuum power flow completed to-date, executed on the leading HPC computing machines available in the NNSA complex today. These preliminary studies indicate relevant pulsed power engineering design simulations can now be completed in (of order) several days, a significant improvement over pre-LDRD levels of performance.

The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead of the standard Laplacian. Recently, it has been established that such a fractional Helmholtz problem better captures the underlying behavior in geophysical electromagnetics. We establish the well-posedness and regularity of this problem. We introduce a hybrid spectral-finite element approach to discretize it and show well-posedness of the discrete system. In addition, we derive a priori discretization error estimates. Finally, we introduce an efficient solver that scales as well as the best possible solver for the classical integer-order Helmholtz equation. We conclude with several illustrative examples that confirm our theoretical findings.

Software development for high-performance scientific computing continues to evolve in response to increased parallelism and the advent of on-node accelerators, in particular GPUs. While these hardware advancements have the potential to significantly reduce turnaround times, they also present implementation and design challenges for engineering codes. We investigate the use of two strategies to mitigate these challenges: the Kokkos library for performance portability across disparate architectures, and the DARMA/vt library for asynchronous many-task scheduling. We investigate the application of Kokkos within the NimbleSM finite element code and the LAMÉ constitutive model library. We explore the performance of DARMA/vt applied to NimbleSM contact mechanics algorithms. Software engineering strategies are discussed, followed by performance analyses of relevant solid mechanics simulations which demonstrate the promise of Kokkos and DARMA/vt for accelerated engineering simulators.

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In this work, we show that reduced communication algorithms for distributed stochastic gradient descent improve the time per epoch and strong scaling for the Generalized Canonical Polyadic (GCP) tensor decomposition, but with a cost, achieving convergence becomes more difficult. The implementation, based on MPI, shows that while one-sided algorithms offer a path to asynchronous execution, the performance benefits of optimized allreduce are difficult to best.

Aeroengines ingest foreign object debris such as sand, which eventually erode components through repeated impacts. Due to the wide feature space, modeling and simulations are needed to rapidly assess the erosion behavior of materials such as composites. Peridynamic simulations were performed to analyze erosion of SiC/SiC composite due to sand impacts, which gives direct insight into the impact erosion mechanism and amounts. The erosion data was strongly correlated to impact velocity and angle, providing predictive equations.

A source-to-source compiler is a type of translator that accepts the source code of a program written in a programming language as its input and produces an equivalent source code in the same or different programming language. S2S techniques are commonly used to enable fluent translation between high-level programming languages, to perform large-scale refactoring operations, and to facilitate instrumentation for dynamic analysis. Negative perceptions about S2S’s applicability in High Performance Computing (HPC) are studied and evaluated here. This is a first study that brings to light reasons why scientists do not use source-to-source techniques for HPC. The primary audience for this paper are those considering S2S technology in their HPC application work.

The accurate construction of a surrogate model is an effective and efficient strategy for performing Uncertainty Quantification (UQ) analyses of expensive and complex engineering systems. Surrogate models are especially powerful whenever the UQ analysis requires the computation of statistics which are difficult and prohibitively expensive to obtain via a direct sampling of the model, e.g. high-order moments and probability density functions. In this paper, we discuss the construction of a polynomial chaos expansion (PCE) surrogate model for radiation transport problems for which quantities of interest are obtained via Monte Carlo simulations. In this context, it is imperative to account for the statistical variability of the simulator as well as the variability associated with the uncertain parameter inputs. More formally, in this paper we focus on understanding the impact of the Monte Carlo transport variability on the recovery of the PCE coefficients. We are able to identify the contribution of both the number of uncertain parameter samples and the number of particle histories simulated per sample in the PCE coefficient recovery. Our theoretical results indicate an accuracy improvement when using few Monte Carlo histories per random sample with respect to configurations with an equivalent computational cost. These theoretical results are numerically illustrated for a simple synthetic example and two configurations of a one-dimensional radiation transport problem in which a slab is represented by means of materials with uncertain cross sections.

To meet the extreme compute demands for deep learning across commercial and scientific applications, dataflow accelerators are becoming increasingly popular. While these “domain-specific” accelerators are not fully programmable like CPUs and GPUs, they retain varying levels of flexibility with respect to data orchestration, i.e., dataflow and tiling optimizations to enhance efficiency. There are several challenges when designing new algorithms and mapping approaches to execute the algorithms for a target problem on new hardware. Previous works have addressed these challenges individually. To address this challenge as a whole, in this work, we present a HW-SW codesign ecosystem for spatial accelerators called Union within the popular MLIR compiler infrastructure. Our framework allows exploring different algorithms and their mappings on several accelerator cost models. Union also includes a plug-and-play library of accelerator cost models and mappers which can easily be extended. The algorithms and accelerator cost models are connected via a novel mapping abstraction that captures the map space of spatial accelerators which can be systematically pruned based on constraints from the hardware, workload, and mapper. We demonstrate the value of Union for the community with several case studies which examine offloading different tensor operations (CONV/GEMM/Tensor Contraction) on diverse accelerator architectures using different mapping schemes.

This work proposes an approach for latent-dynamics learning that exactly enforces physical conservation laws. The method comprises two steps. First, the method computes a low-dimensional embedding of the high-dimensional dynamical-system state using deep convolutional autoencoders. This defines a low-dimensional nonlinear manifold on which the state is subsequently enforced to evolve. Second, the method defines a latent-dynamics model that associates with the solution to a constrained optimization problem. Here, the objective function is defined as the sum of squares of conservation-law violations over control volumes within a finite-volume discretization of the problem; nonlinear equality constraints explicitly enforce conservation over prescribed subdomains of the problem. Under modest conditions, the resulting dynamics model guarantees that the time-evolution of the latent state exactly satisfies conservation laws over the prescribed subdomains.

Boolean functions and binary arithmetic operations are central to standard computing paradigms. Accordingly, many advances in computing have focused upon how to make these operations more efficient as well as exploring what they can compute. To best leverage the advantages of novel computing paradigms it is important to consider what unique computing approaches they offer. However, for any special-purpose co-processor, Boolean functions and binary arithmetic operations are useful for, among other things, avoiding unnecessary I/O on-and-off the co-processor by pre- and post-processing data on-device. This is especially true for spiking neuromorphic architectures where these basic operations are not fundamental low-level operations. Instead, these functions require specific implementation. Here we discuss the implications of an advantageous streaming binary encoding method as well as a handful of circuits designed to exactly compute elementary Boolean and binary operations.

This report presents the results of a collaborative effort under the Verification, Validation, and Uncertainty Quantification (VVUQ) thrust area of the North American Energy Resilience Model (NAERM) program. The goal of the effort described in this report was to integrate the Dakota software with the NAERM software framework to demonstrate sensitivity analysis of a co-simulation for NAERM.

In this paper, we introduce and analyze a new class of optimal control problems constrained by elliptic equations with uncertain fractional exponents. We utilize risk measures to formulate the resulting optimization problem. We develop a functional analytic framework, study the existence of solution, and rigorously derive the first-order optimality conditions. Additionally, we employ a sample-based approximation for the uncertain exponent and the finite element method to discretize in space. We prove the rate of convergence for the optimal risk neutral controls when using quadrature approximation for the uncertain exponent and conclude with illustrative examples.

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Process-structure linkage is one of the most important topics in materials science due to the fact that virtually all information related to the materials, including manufacturing processes, lies in the microstructure itself. Therefore, to learn more about the process, one must start by thoroughly examining the microstructure. This gives rise to inverse problems in the context of process-structure linkages, which attempt to identify the processes that were used to manufacturing the given microstructure. In this work, we present an inverse problem for structure-process linkages which we solve using asynchronous parallel Bayesian optimization which exploits parallel computing resources. We demonstrate the effectiveness of the method using kinetic Monte Carlo model for grain growth simulation.

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We present StressBench, a network benchmarking framework written for testing MPI operations and file I/O concurrently. It is designed specifically to execute MPI communication and file access patterns that are representative of real-world scientific applications. Existing tools consider either the worst case congestion with small abstract patterns or peak performance with simplistic patterns. StressBench allows for a richer study of congestion by allowing orchestration of network load scenarios that are representative of those typically seen at HPC centres, something that is difficult to achieve with existing tools. We demonstrate the versatility of the framework from micro benchmarks through to finely controlled congested runs across a cluster. Validation of the results using four proxy application communication schemes within StressBench against parent applications shows a maximum difference of 15%. Using the I/O modeling capabilities of StressBench, we are able to quantify the impact of file I/O on application traffic showing how it can be used in procurement and performance studies.