Bayesian inference with a simple Gaussian error model is used to efficiently compute prediction variances for energies, forces, and stresses in the linear SNAP interatomic potential. The prediction variance is shown to have a strong correlation with the absolute error over approximately 24 orders of magnitude. Using this prediction variance, an active learning algorithm is constructed to iteratively train a potential by selecting the structures with the most uncertain properties from a pool of candidate structures. The relative importance of the energy, force, and stress errors in the objective function is shown to have a strong impact upon the trajectory of their respective net error metrics when running the active learning algorithm. Batched training of different batch sizes is also tested against singular structure updates, and it is found that batches can be used to significantly reduce the number of retraining steps required with only minor impact on the active learning trajectory.
Computational singular perturbation (CSP) is a method to analyze dynamical systems. It targets the decoupling of fast and slow dynamics using an alternate linear expansion of the right-hand side of the governing equations based on eigenanalysis of the associated Jacobian matrix. This representation facilitates diagnostic analysis, detection and control of stiffness, and the development of simplified models. We have implemented CSP in a C++ open-source library CSPlib1 using the Kokkos2 parallel programming model to address portability across diverse heterogeneous computing platforms, i.e., multi/many-core CPUs and GPUs. We describe the CSPlib implementation and present its computational performance across different computing platforms using several test problems. Specifically, we test the CSPlib performance for a constant pressure ignition reactor model on different architectures, including IBM Power 9, Intel Xeon Skylake, and NVIDIA V100 GPU. The size of the chemical kinetic mechanism is varied in these tests. As expected, the Jacobian matrix evaluation, the eigensolution of the Jacobian matrix, and matrix inversion are the most expensive computational tasks. When considering the higher throughput characteristic of GPUs, GPUs performs better for small matrices with higher occupancy rate. CPUs gain more advantages from the higher performance of well-tuned and optimized linear algebra libraries such as OpenBLAS. Program summary: Program Title: CSPlib CPC Library link to program files: https://doi.org/10.17632/p9gb7z54sp.1 Developer's repository link: https://github.com/sandialabs/csplib Licensing provisions: BSD 2-clause Programming language: C++ Nature of problem: Dynamical systems can involve coupled processes with a wide range of time scales. The computational singular perturbation (CSP) method offers a reformulation of these systems which enables the use of dynamically-based diagnostic tools to better comprehend the dynamics by decoupling fast and slow processes. CSPlib is an open-source software library for analyzing general ordinary differential equation (ODE) and differential algebraic equation (DAE) systems, with specialized implementations for detailed chemical kinetic ODE/DAE systems. It relies on CSP for the analysis of these systems. CSPlib has been used in gas kinetic and heterogeneous catalytic kinetic models. Solution method: CSP analysis seeks a set of basis vectors to linearly decompose the right-hand side (RHS) of a dynamical system in a manner that decouples fast and slow processes. The CSP basis vectors are often well approximated with the right eigenvectors of the RHS Jacobian. And the left basis vectors are found by the inversion of the matrix, whose columns are the CSP basis vectors. Accordingly, the right and left CSP basis vectors are orthonormal. CSP defines mode amplitudes as the projections of the left basis vectors on the RHS; the time scales as the reciprocals of the RHS Jacobian eigenvalue magnitudes; and the CSP pointers, which are the element-wise multiplication of the transpose of the right CSP basis vectors with the left CSP basis vectors. For kinetic models that can be cast as the product of a generalized stoichiometric matrix and a rate of progress vector, CSP defines the participation index, which represents the contribution of a chemical reaction to each mode. Further, it defines the slow and fast importance indices, which describe the contribution of a chemical reaction to the slow and fast dynamics of a state variable, respectively. These indices are useful in diagnostic studies of dynamical systems and the construction of simplified models. Additional comments including restrictions and unusual features: CSPlib is a portable library that carries out many CSP analyses in parallel and can be used in modern high-performance platforms.
Characterizing and quantifying microstructure evolution is critical to forming quantitative relationships between material processing conditions, resulting microstructure, and observed properties. Machine-learning methods are increasingly accelerating the development of these relationships by treating microstructure evolution as a pattern recognition problem, discovering relationships explicitly or implicitly. These methods often rely on identifying low-dimensional microstructural fingerprints as latent variables. However, using inappropriate latent variables can lead to challenges in learning meaningful relationships. In this work, we survey and discuss the ability of various linear and nonlinear dimensionality reduction methods including principal component analysis, autoencoders, and diffusion maps to quantify and characterize the learned latent space microstructural representations and their time evolution. We characterize latent spaces by their ability to represent high-dimensional microstructural data in terms of compression achieved as a function of the number of latent dimensions required to represent the data accurately, their accuracy based on their reconstruction performance, and the smoothness of the microstructural trajectories in latent dimension. We quantify these metrics for common microstructure evolution problems in material science including spinodal decomposition of a binary metallic alloy, thin film deposition of a binary metallic alloy, dendritic growth, and grain growth in a polycrystal. This study provides considerations and guidelines for choosing dimensionality reduction methods when considering materials problems that involve high dimensional data and a variety of features over a range of lengths and time scales.
This paper focuses on the development of multifidelity modeling approaches using neural network surrogates, where training data arising from multiple model forms and resolutions are integrated to predict high-fidelity response quantities of interest at lower cost. We focus on the context of quantum chemistry and the integration of information from multiple levels of theory. Important foundations include the use of symmetry function-based atomic energy vector constructions as feature vectors for representing structures across families of molecules and single-fidelity neural network training capabilities that learn the relationships needed to map feature vectors to potential energy predictions. These foundations are embedded within several multifidelity topologies that decompose the high-fidelity mapping into model-based components, including sequential formulations that admit a general nonlinear mapping across fidelities and discrepancy-based formulations that presume an additive decomposition. Methodologies are first explored and demonstrated on a pair of simple analytical test problems and then deployed for potential energy prediction for C5H5 using B2PLYP-D3/6-311++G(d,p) for high-fidelity simulation data and Hartree-Fock 6-31G for low-fidelity data. For the common case of limited access to high-fidelity data, our computational results demonstrate that multifidelity neural network potential energy surface constructions achieve roughly an order of magnitude improvement, either in terms of test error reduction for equivalent total simulation cost or reduction in total cost for equivalent error.
The automated kinetics workflow code, KinBot, was used to explore and characterize the regions of the C7H7 potential energy surface that are relevant to combustion environments and especially soot inception. We first explored the lowest-energy region, which includes the benzyl, fulvenallene + H, and cyclopentadienyl + acetylene entry points. We then expanded the model to include two higher-energy entry points, vinylpropargyl + acetylene and vinylacetylene + propargyl. The automated search was able to uncover the pathways from the literature. In addition, three important new routes were discovered: a lower-energy pathway connecting benzyl with vinylcyclopentadienyl, a decomposition mechanism from benzyl that results in side-chain hydrogen atom loss to produce fulvenallene + H, and shorter and lower energy routes to the dimethylene-cyclopentenyl intermediates. We systematically reduced the extended model to a chemically relevant domain composed of 63 wells, 10 bimolecular products, 87 barriers, and 1 barrierless channel and constructed a master equation using the CCSD(T)-F12a/cc-pVTZ//ωB97X-D/6-311++G(d,p) level of theory to provide rate coefficients for chemical modeling. Our calculated rate coefficients show excellent agreement with measured ones. We also simulated concentration profiles and calculated branching fractions from the important entry points to provide an interpretation of this important chemical landscape.
Automation of rate-coefficient calculations for gas-phase organic species became possible in recent years and has transformed how we explore these complicated systems computationally. Kinetics workflow tools bring rigor and speed and eliminate a large fraction of manual labor and related error sources. In this paper we give an overview of this quickly evolving field and illustrate, through five detailed examples, the capabilities of our own automated tool, KinBot. We bring examples from combustion and atmospheric chemistry of C-, H-, O-, and N-atom-containing species that are relevant to molecular weight growth and autoxidation processes. The examples shed light on the capabilities of automation and also highlight particular challenges associated with the various chemical systems that need to be addressed in future work.
Neural ordinary differential equations (NODEs) have recently regained popularity as large-depth limits of a large class of neural networks. In particular, residual neural networks (ResNets) are equivalent to an explicit Euler discretization of an underlying NODE, where the transition from one layer to the next is one time step of the discretization. The relationship between continuous and discrete neural networks has been of particular interest. Notably, analysis from the ordinary differential equation viewpoint can potentially lead to new insights for understanding the behavior of neural networks in general. In this work, we take inspiration from differential equations to define the concept of stiffness for a ResNet via the interpretation of a ResNet as the discretization of a NODE. Here, we then examine the effects of stiffness on the ability of a ResNet to generalize, via computational studies on example problems coming from climate and chemistry models. We find that penalizing stiffness does have a unique regularizing effect, but we see no benefit to penalizing stiffness over L2 regularization (penalization of network parameter norms) in terms of predictive performance.
Numerical algorithms for stiff stochastic differential equations are developed using linear approximations of the fast diffusion processes, under the assumption of decoupling between fast and slow processes. Three numerical schemes are proposed, all of which are based on the linearized formulation albeit with different degrees of approximation. The schemes are of comparable complexity to the classical explicit Euler-Maruyama scheme but can achieve better accuracy at larger time steps in stiff systems. Convergence analysis is conducted for one of the schemes, that shows it to have a strong convergence order of 1/2 and a weak convergence order of 1. Approximations arriving at the other two schemes are discussed. Numerical experiments are carried out to examine the convergence of the schemes proposed on model problems.
Nonlocal models provide a much-needed predictive capability for important Sandia mission applications, ranging from fracture mechanics for nuclear components to subsurface flow for nuclear waste disposal, where traditional partial differential equations (PDEs) models fail to capture effects due to long-range forces at the microscale and mesoscale. However, utilization of this capability is seriously compromised by the lack of a rigorous nonlocal interface theory, required for both application and efficient solution of nonlocal models. To unlock the full potential of nonlocal modeling we developed a mathematically rigorous and physically consistent interface theory and demonstrate its scope in mission-relevant exemplar problems.
This article illustrates the use of unsupervised probabilistic learning techniques for the analysis of planetary reentry trajectories. A three-degree-of-freedom model was employed to generate optimal trajectories that comprise the training datasets. The algorithm first extracts the intrinsic structure in the data via a diffusion map approach. We find that data resides on manifolds of much lower dimensionality compared to the high-dimensional state space that describes each trajectory. Using the diffusion coordinates on the graph of training samples, the probabilistic framework subsequently augments the original data with samples that are statistically consistent with the original set. The augmented samples are then used to construct conditional statistics that are ultimately assembled in a path planning algorithm. In this framework, the controls are determined stage by stage during the flight to adapt to changing mission objectives in real-Time.