Publications

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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.

In this paper we introduce EMPIRE-PIC, a finite element method particle-in-cell (FEM-PIC) application developed at Sandia National Laboratories. The code has been developed in C++ using the Trilinos library and the Kokkos Performance Portability Framework to enable running on multiple modern compute architectures while only requiring maintenance of a single codebase. EMPIRE-PIC is capable of solving both electrostatic and electromagnetic problems in two- and three-dimensions to second-order accuracy in space and time. In this paper we validate the code against three benchmark problems - a simple electron orbit, an electrostatic Langmuir wave, and a transverse electromagnetic wave propagating through a plasma. We demonstrate the performance of EMPIRE-PIC on four different architectures: Intel Haswell CPUs, Intel's Xeon Phi Knights Landing, ARM Thunder-X2 CPUs, and NVIDIA Tesla V100 GPUs attached to IBM POWER9 processors. This analysis demonstrates scalability of the code up to more than two thousand GPUs, and greater than one hundred thousand CPUs.

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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The application of deep learning toward discovery of data-driven models requires careful application of inductive biases to obtain a description of physics which is both accurate and robust. We present here a framework for discovering continuum models from high fidelity molecular simulation data. Our approach applies a neural network parameterization of governing physics in modal space, allowing a characterization of differential operators while providing structure which may be used to impose biases related to symmetry, isotropy, and conservation form. Here, we demonstrate the effectiveness of our framework for a variety of physics, including local and nonlocal diffusion processes and single and multiphase flows. For the flow physics we demonstrate this approach leads to a learned operator that generalizes to system characteristics not included in the training sets, such as variable particle sizes, densities, and concentration.

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Machine learning and artificial intelligence (ML/AI) methods have been used successfully in recent years to solve problems in many areas, including image recognition, unsupervised and supervised classification, game-playing, system identification and prediction, and autonomous vehicle control. Data-driven machine learning methods have also been applied to fusion energy research for over 2 decades, including significant advances in the areas of disruption prediction, surrogate model generation, and experimental planning. The advent of powerful and dedicated computers specialized for large-scale parallel computation, as well as advances in statistical inference algorithms, have greatly enhanced the capabilities of these computational approaches to extract scientific knowledge and bridge gaps between theoretical models and practical implementations. Large-scale commercial success of various ML/AI applications in recent years, including robotics, industrial processes, online image recognition, financial system prediction, and autonomous vehicles, have further demonstrated the potential for data-driven methods to produce dramatic transformations in many fields. These advances, along with the urgency of need to bridge key gaps in knowledge for design and operation of reactors such as ITER, have driven planned expansion of efforts in ML/AI within the US government and around the world. The Department of Energy (DOE) Office of Science programs in Fusion Energy Sciences (FES) and Advanced Scientific Computing Research (ASCR) have organized several activities to identify best strategies and approaches for applying ML/AI methods to fusion energy research. This paper describes the results of a joint FES/ASCR DOE-sponsored Research Needs Workshop on Advancing Fusion with Machine Learning, held April 30–May 2, 2019, in Gaithersburg, MD (full report available at https://science.osti.gov/-/media/fes/pdf/workshop-reports/FES_ASCR_Machine_Learning_Report.pdf). The workshop drew on broad representation from both FES and ASCR scientific communities, and identified seven Priority Research Opportunities (PRO’s) with high potential for advancing fusion energy. In addition to the PRO topics themselves, the workshop identified research guidelines to maximize the effectiveness of ML/AI methods in fusion energy science, which include focusing on uncertainty quantification, methods for quantifying regions of validity of models and algorithms, and applying highly integrated teams of ML/AI mathematicians, computer scientists, and fusion energy scientists with domain expertise in the relevant areas.

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In this work, a stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm. This involves adding some artificial diffusion to the high order, semi-discrete method and mass lumping in the time derivative term. The result is a low order method that provides local extremum diminishing properties for hyperbolic systems. The difference between the low order method and the high order method is scaled element-wise using a limiter and added to the low order scheme. The limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy. The stabilization also involves an optional consistent background high order dissipation that reduces phase errors. The resulting stabilized scheme is a semi-discrete method that can be applied to inviscid shock MHD problems and may be even extended to resistive and viscous MHD problems. To satisfy the divergence free constraint of the MHD equations, we add parabolic divergence cleaning to the system. Various time integration methods can be used to discretize the scheme in time. We demonstrate the robustness of the scheme by solving several shock MHD problems.

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This report reviews a hierarchy of formal mathematical models for describing plasma phenomena. Starting with the Boltzmann equation, a sequence of approximations and modeling assumptions can be made that progressively reduce to the equations for magnetohydrodynamics. Understanding the assumptions behind each of these models and their mathematical form is essential to appropriate use of each level of the hierarchy. A sequence of moment models of the Boltzmann equation are presented, then focused into a generalized three-fluid model for neutral species, electrons, and ions. This model is then further reduced to a two-fluid model, for which Braginskii described a useful closure. Further reduction of the two-fluid model yields a Generalized Ohm's Law model, which provides a connection to magnetohydrodynamic approaches. A verification approach based on linear plasma waves is presented alongside the model hierarchy, which is intended as an initial and necessary but not sufficient step for verification of plasma models within this hierarchy.

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Residual neural networks (ResNets) are a promising class of deep neural networks that have shown excellent performance for a number of learning tasks, e.g., image classification and recognition. Mathematically, ResNet architectures can be interpreted as forward Euler discretizations of a nonlinear initial value problem whose time-dependent control variables represent the weights of the neural network. Hence, training a ResNet can be cast as an optimal control problem of the associated dynamical system. For similar time-dependent optimal control problems arising in engineering applications, parallel-in-time methods have shown notable improvements in scalability. This paper demonstrates the use of those techniques for efficient and effective training of ResNets. The proposed algorithms replace the classical (sequential) forward and backward propagation through the network layers with a parallel nonlinear multigrid iteration applied to the layer domain. This adds a new dimension of parallelism across layers that is attractive when training very deep networks. From this basic idea, we derive multiple layer-parallel methods. The most efficient version employs a simultaneous optimization approach where updates to the network parameters are based on inexact gradient information in order to speed up the training process. Finally, using numerical examples from supervised classification, we demonstrate that the new approach achieves a training performance similar to that of traditional methods, but enables layer-parallelism and thus provides speedup over layer-serial methods through greater concurrency.