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DOE OSTI2023

Robust scalable initialization for Bayesian variational inference with multi-modal Laplace approximations

Predictive modeling typically relies on Bayesian model calibration to provide uncertainty quantification. Variational inference utilizing fully independent (“mean-field”) Gaussian distributions are often used as approximate probability density functions. This simplification is attractive since the number of variational parameters grows only linearly with the number of unknown model parameters. However, the resulting diagonal covariance structure and unimodal behavior can be too restrictive to provide useful approximations of intractable Bayesian posteriors that exhibit highly non-Gaussian behavior, including multimodality. High-fidelity surrogate posteriors for these problems can be obtained by considering the family of Gaussian mixtures. Gaussian mixtures are capable of capturing multiple modes and approximating any distribution to an arbitrary degree of accuracy, while maintaining some analytical tractability. Unfortunately, variational inference using Gaussian mixtures with full-covariance structures suffers from a quadratic growth in variational parameters with the number of model parameters. The existence of multiple local minima due to strong nonconvex trends in the loss functions often associated with variational inference present additional complications, These challenges motivate the need for robust initialization procedures to improve the performance and computational scalability of variational inference with mixture models. In this work, we propose a method for constructing an initial Gaussian mixture model approximation that can be used to warm-start the iterative solvers for variational inference. The procedure begins with a global optimization stage in model parameter space. In this step, local gradient-based optimization, globalized through multistart, is used to determine a set of local maxima, which we take to approximate the mixture component centers. Around each mode, a local Gaussian approximation is constructed via the Laplace approximation. Finally, the mixture weights are determined through constrained least squares regression. The robustness and scalability of the proposed methodology is demonstrated through application to an ensemble of synthetic tests using high-dimensional, multimodal probability density functions. Here, the practical aspects of the approach are demonstrated with inversion problems in structural dynamics.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Fault geometry and slip distribution of the 2008 M w 7.9 Wenchuan, China earthquake, inferred from GPS and InSAR measurements

We revisit the problem of coseismic rupture of the 2008 M w 7.9 Wenchuan earthquake. Precise determination of the fault structure and slip distribution provides critical information about the mechanical behaviour of the fault system and earthquake rupture. We use all the geodetic data available, craft a more realistic Earth structure and fault model compared to previous studies, and employ a nonlinear inversion scheme to optimally solve for the fault geometry and slip distribution. Compared to a homogeneous elastic half-space model and laterally uniform layered models, adopting separate layered elastic structure models on both sides of the Beichuan fault significantly improved data fitting. Our results reveal that: (1) The Beichuan fault is listric in shape, with near surface fault dip angles increasing from ~36° at the southwest end to ~83° at the northeast end of the rupture. (2) The fault rupture style changes from predominantly thrust at the southwest end to dextral at the northeast end of the fault rupture. (3) Fault slip peaks near the surface for most parts of the fault, with~8.4mthrust and ~5 m dextral slip near Hongkou and ~6 m thrust and ~8.4 m dextral slip near Beichuan, respectively. (4) The peak slips are located around fault geometric complexities, suggesting that earthquake style and rupture propagation were determined by fault zone geometric barriers. Such barriers exist primarily along restraining left stepping discontinuities of the dextralcompressional fault system. (5) The seismic moment released on the fault above 20 km depth is 8.2 × 10 21 N m, corresponding to an M w 7.9 event. The seismic moments released on the local slip concentrations are equivalent to events of M w 7.5 at Yingxiu-Hongkou, M w 7.3 at Beichuan-Pingtong, M w 7.2 near Qingping, M w 7.1 near Qingchuan, and M w 6.7 near Nanba, respectively. (6) The fault geometry and kinematics are consistent with a model in which crustal deformation at the eastern margin of the Tibetan plateau is decoupled by differential motion across a decollement in the mid crust, above which deformation is dominated by brittle reverse faulting and below which deformation occurs by viscous horizontal shortening and vertical thickening.

58 GEOSCIENCES↗

DOE OSTI2021

Initial use of Nek5000/Cardinal to improve closure models in Pronghorn

Heat transfer coefficient closure models for pebble bed reactors are built using a data-driven approach by leveraging the capabilities of an Evolutionary Algorithm entitled Particle Swarm Optimization (PSO). In the present work, the Computational Fluid Dynamics code nekRS was used in order to collect the high-fidelity flow data for a core with 1,568 pebbles. To characterize the heat transfer, multiple concentric regions were considered to extract the physical quantities of interest, e.g./ the Reynolds number. The PSO algorithm is employed as part of an inverse problem targeting determine what are the coefficients for a Nusselt number correlation to match the collected data. Such correlation should follow any given format that is defined a priori. Finally, two correlations are proposed, one with an implicit dependence on the pebbles’ wall temperatures and another expressed as a fully explicit correlation depending on the flow conditions and the position within the core. Anyway, given the generic nature of the proposed approach, correlations following different formats could be tested. Preliminary results for the high-fidelity simulation of a fast MSR core are presented. The target Reynolds number is currently 20K, with the expectation that this will increase, pending the availability of further computational resources. These simulations will be used to inform lower fidelity models, including a coarse CFD turbulence model in Pronghorn. Additionally, they will serve as a reference for the RANS models in Nek5000/NekRS.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2019

Efficient optimal reconstruction of linear fields and band-powers from cosmological data

We present an efficient implementation of Wiener filtering of real-space linear field and optimal quadratic estimator of its power spectrum Band-powers. We first recast the field reconstruction into an optimization problem, which we solve using quasi-Newton optimization. We then recast the power spectrum estimation into the field marginalization problem, from which we obtain an expression that depends on the field reconstruction solution and a determinant term. We develop a novel simulation based method for the latter. We extend the simulations formalism to provide the covariance matrix for the power spectrum. We develop a flexible framework that can be used on a variety of cosmological fields and present results for a variety of test cases, using simulated examples of projected density fields, projected shear maps from galaxy lensing, and observed Cosmic Microwave Background (CMB) temperature anisotropies, with a wide range of map incompleteness and variable noise. For smaller cases where direct numerical inversion is possible, we show that our solution matches that created by direct Wiener Filtering at a fraction of the overall computation cost. Even more significant reduction of computational is achieved by this implementation of optimal quadratic estimator due to the fast evaluation of the Hessian matrix. This technique allows for accurate map and power spectrum reconstruction with complex masks and nontrivial noise properties.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

Projected Multi-Agent Consensus Equilibrium (PMACE) With Application to Ptychography

Multi-Agent Consensus Equilibrium (MACE) formulates an inverse imaging problem as a balance among multiple update agents such as data-fitting terms and denoisers. However, each such agent operates on a separate copy of the full image, leading to redundant memory use and slow convergence when each agent affects only a small subset of the full image. In this article, we extend MACE to Projected Multi-Agent Consensus Equilibrium (PMACE), in which each agent updates only a projected component of the full image, thus greatly reducing memory use for some applications. We describe PMACE in terms of an equilibrium problem and an equivalent fixed-point problem and show that in most cases the PMACE equilibrium is not the solution of an optimization problem. To demonstrate the value of PMACE, we apply it to the problem of ptychography, in which a sample is reconstructed from the diffraction patterns resulting from coherent X-ray illumination at multiple overlapping spots. In our PMACE formulation, each spot corresponds to a separate data-fitting agent, with the final solution found as an equilibrium among all the agents. In conclusion, our results demonstrate that the PMACE reconstruction algorithm generates more accurate reconstructions at a lower computational cost than existing ptychography algorithms when the spots are sparsely sampled.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Adaptive hyperparameter updating for training restricted Boltzmann machines on quantum annealers

Restricted Boltzmann Machines (RBMs) have been proposed for developing neural networks for a variety of unsupervised machine learning applications such as image recognition, drug discovery, and materials design. The Boltzmann probability distribution is used as a model to identify network parameters by optimizing the likelihood of predicting an output given hidden states trained on available data. Training such networks often requires sampling over a large probability space that must be approximated during gradient based optimization. Quantum annealing has been proposed as a means to search this space more efficiently which has been experimentally investigated on D-Wave hardware. D-Wave implementation requires selection of an effective inverse temperature or hyperparameter (β) within the Boltzmann distribution which can strongly influence optimization. Here, we show how this parameter can be estimated as a hyperparameter applied to D-Wave hardware during neural network training by maximizing the likelihood or minimizing the Shannon entropy. We find both methods improve training RBMs based upon D-Wave hardware experimental validation on an image recognition problem. Neural network image reconstruction errors are evaluated using Bayesian uncertainty analysis which illustrate more than an order magnitude lower image reconstruction error using the maximum likelihood over manually optimizing the hyperparameter. The maximum likelihood method is also shown to out-perform minimizing the Shannon entropy for image reconstruction.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Scaling quantum approximate optimization on near-term hardware

The quantum approximate optimization algorithm (QAOA) is an approach for near-term quantum computers to potentially demonstrate computational advantage in solving combinatorial optimization problems. However, the viability of the QAOA depends on how its performance and resource requirements scale with problem size and complexity for realistic hardware implementations. Here, we quantify scaling of the expected resource requirements by synthesizing optimized circuits for hardware architectures with varying levels of connectivity. Assuming noisy gate operations, we estimate the number of measurements needed to sample the output of the idealized QAOA circuit with high probability. We show the number of measurements, and hence total time to solution, grows exponentially in problem size and problem graph degree as well as depth of the QAOA ansatz, gate infidelities, and inverse hardware graph degree. These problems may be alleviated by increasing hardware connectivity or by recently proposed modifications to the QAOA that achieve higher performance with fewer circuit layers.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

NSGA-PINN: A Multi-Objective Optimization Method for Physics-Informed Neural Network Training

This paper presents NSGA-PINN, a multi-objective optimization framework for the effective training of physics-informed neural networks (PINNs). The proposed framework uses the non-dominated sorting genetic algorithm (NSGA-II) to enable traditional stochastic gradient optimization algorithms (e.g., ADAM) to escape local minima effectively. Additionally, the NSGA-II algorithm enables satisfying the initial and boundary conditions encoded into the loss function during physics-informed training precisely. We demonstrate the effectiveness of our framework by applying NSGA-PINN to several ordinary and partial differential equation problems. In particular, we show that the proposed framework can handle challenging inverse problems with noisy data.

Lu, Binghang (ORCID:0009000160016632)↗

DOE OSTI2021

Feasibility of DEIM for retrieving the initial field via dimensionality reduction

When parameter estimation is solved in a high-dimensional space, the dimensionality reduction strategy becomes the primary consideration for alleviating the tremendous computational cost. Here, the discrete empirical interpolation method (DEIM) is explored to retrieve the initial condition (IC) by combining the polynomial chaos (PC) based ensemble Kalman filter (i.e. PC-EnKF), where a non-intrusive PC expansion is considered as a surrogate model in place of the forward model in the prediction step of the ensemble Kalman filter, resulting in fewer forward model integrations but with a comparable accuracy as Monte Carlo-based approaches. The DEIM acts as a hyper-reduction tool to provide the low-dimensional input for the high-dimensional initial field, which can be reconstructed using the information on the sparse interpolation grid points that is adaptively obtained through PC-EnKF data assimilation method. Thus an innovative framework to reconstruct the IC is developed. The detailed procedure at each assimilation iteration includes: the determination of the spatial interpolation points, the estimation of the initial values on the interpolation locations using the optimal observations, and the reconstruction of IC in the full space. The current study uses the reconstruction field of initial conditions of the Navier-Stokes equations as an example to illustrate the efficacy of our method. The experimental results demonstrate the proposed algorithm achieves a satisfactory reconstruction for the initial field. The proposed method helps to extend the applicable area of DEIM in solving inverse problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness

We report the accuracy of deep learning, i.e., deep neural networks, can be characterized by dividing the total error into three main types: approximation error, optimization error, and generalization error. Whereas there are some satisfactory answers to the problems of approximation and optimization, much less is known about the theory of generalization. Most existing theoretical works for generalization fail to explain the performance of neural networks in practice. To derive a meaningful bound, we study the generalization error of neural networks for classification problems in terms of data distribution and neural network smoothness. We introduce the cover complexity (CC) to measure the difficulty of learning a data set and the inverse of the modulus of continuity to quantify neural network smoothness. A quantitative bound for expected accuracy/error is derived by considering both the CC and neural network smoothness. Although most of the analysis is general and not specific to neural networks, we validate our theoretical assumptions and results numerically for neural networks by several data sets of images. The numerical results confirm that the expected error of trained networks scaled with the square root of the number of classes has a linear relationship with respect to the CC. We also observe a clear consistency between test loss and neural network smoothness during the training process. In addition, we demonstrate empirically that the neural network smoothness decreases when the network size increases whereas the smoothness is insensitive to training dataset size.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Measurement of a radial flow profile with eddy current flow meters and deep neural networks

Eddy current flow meters (ECFMs) measure flows of conductive fluids. Recent interest in ECFMs has increased due to applications in advanced nuclear reactors. ECFMs are well suited for such applications, as they can provide non-invasive measurements of flow in fluids that are often difficult to measure. Traditionally, ECFMs are operated using an alternating current at a single frequency, limiting ECFMs to measure average fluid velocities, blockages, or voids. Here, we expand the capabilities of ECFMs by measuring the fluid radial velocity profile of liquid mercury. To accomplish this, we made several ECFM sensitivity measurements at a range of frequencies. Different frequencies vary the electromagnetic skin depth of the device. By adjusting frequencies, we probed the fluid velocity at various radial locations and constructed a flow-velocity profile. The relationship between the ECFM measurements and velocity profile is nonlinear and requires solving an inverse problem. Using electromagnetic finite-element simulations to train a deep neural network (DNN), we created a model that provides a stable general relationship between the sensitivity measurements of an ECFM and the fluid velocity profile. Using ECFM measurements of liquid mercury, our DNN model calculates a flow profile that agrees well with computational fluid dynamics (CFD) simulations. This technique has potential to improve flow monitoring for optimization, safe operation of conductive fluid loops, and/or validating complex CFD models.

47 OTHER INSTRUMENTATION↗

DOE OSTI2019

Highly-scalable, physics-informed GANs for learning solutions of stochastic PDEs

Uncertainty quantification for forward and inverse problems is a central challenge across physical and biomedical disciplines. We address this challenge for the problem of modeling subsurface flow at the Hanford Site by combining stochastic computational models with observational data using physics-informed GAN models. The geographic extent, spatial heterogeneity, and multiple correlation length scales of the Hanford Site require training a computationally intensive GAN model to thousands of dimensions. We develop a highly optimized implementation that scales to {\herogpucount} NVIDIA Volta GPUs. We develop a hierarchical scheme based on a multi-player game-theoretic approach for exploiting domain parallelism, map discriminators and generators to multiple GPUs, and employ efficient communication schemes to ensure training stability and convergence. Our implementation scales to {\heronodecount} nodes on the Summit supercomputer with a \weakscalingeff\% scaling efficiency, achieving peak and sustained half-precision rates of {\heropeakperf} PF/s and {\herosustainedperf} PF/s.

physics informed ML, machine learning, generative ↗

DOE OSTI2018

Adjoint-based Calibration of Plasticity Model Parameters from Digital Image Correlation Data

Parameter estimation for mechanical models of plastic deformation utilized in nuclear weapons systems is a laborious process for both experimentalists and constitutive modelers and is critical to producing meaningful numerical predictions. In this work we derive an adjoint-based optimization approach for a stabilized, large-deformation J 2 plasticity model that is considerably more computationally efficient but no less accurate than current state of the art methods. Unlike most approaches to model calibration, we drive the inversion procedure with full-field deformation data that can be experimentally measured through established digital image or volume correlation techniques. We present numerical results for two and three dimensional model problems and comment on various directions of future research.

98 NUCLEAR DISARMAMENT, SAFEGUARDS, AND PHYSICAL P↗

DOE OSTI2024

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ-approximate inverse Cholesky factors of the kernel matrices with complexity O(N log d (N/ϵ)) in space and O(N log 2d (N/ϵ)) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

fPINNs: Fractional Physics-Informed Neural Networks

Physics-informed neural networks (PINNs), introduced, are effective in solving integer-order partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedforward neural networks (NNs) with the PDEs explicitly encoded into the NN using automatic differentiation, while the sum of the mean-squared PDE residuals and the mean-squared error in initial-boundary conditions is minimized with respect to the NN parameters. Here we extend PINNs to fractional PINNs (fPINNs) to solve space-time fractional advection-diffusion equations (fractional ADEs), and we study systematically their convergence, hence explaining both fPINNs and PINNs for the first time. Specifically, we demonstrate their accuracy and effectiveness in solving multidimensional forward and inverse problems with forcing terms whose values are only known at randomly scattered spatio-temporal coordinates (black-box (BB) forcing terms). A novel element of the fPINNs is the hybrid approach that we introduce for constructing the residual in the loss function using both automatic differentiation for the integer-order operators and numerical discretization for the fractional operators. This approach bypasses the difficulties stemming from the fact that automatic differentiation is not applicable to fractional operators because the standard chain rule in integer calculus is not valid in fractional calculus. To discretize the fractional operators, we employ the Grünwald--Letnikov (GL) formula in one-dimensional fractional ADEs and the vector GL formula in conjunction with the directional fractional Laplacian in two- and three-dimensional fractional ADEs. We first consider the one-dimensional fractional Poisson equation and compare the convergence of the fPINNs against the finite difference method (FDM). We present the solution convergence using both the mean L 2 error as well as the standard deviation due to sensitivity to NN parameter initializations. Using different GL formulas we observe first-, second-, and third-order convergence rates for small size training sets but the error saturates for larger training sets. We explain these results by analyzing the four sources of numerical errors due to discretization, sampling, NN approximation, and optimization. The total error decays monotonically (below 10 –5 for a third-order GL formula) but it saturates beyond that point due to the optimization error. We also analyze the relative balance between discretization and sampling errors and observe that the sampling size and the number of discretization points (auxiliary points) should be comparable to achieve the highest accuracy. As we increase the depth of the NN up to certain value, the mean error decreases and the standard deviation increases, whereas the width has essentially no effect unless its value is either too small or too large. We next consider time-dependent fractional ADEs and compare white-box (WB) and BB forcing. We observe that for the WB forcing, our results are similar to the aforementioned cases; however, for the BB forcing fPINNs outperform FDM. Subsequently, we consider multidimensional time-, space-, and space-time-fractional ADEs using the directional fractional Laplacian and we observe relative errors of 10 –3 ~ 10 –4 . Lastly, we solve several inverse problems in one, two, and three dimensions to identify the fractional orders, diffusion coefficients, and transport velocities and obtain accurate results given proper initializations even in the presence of significant noise.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

On the Solution of ℓ 0 -Constrained Sparse Inverse Covariance Estimation Problems

The sparse inverse covariance matrix is used to model conditional dependencies between variables in a graphical model to fit a multivariate Gaussian distribution. Estimating the matrix from data are well known to be computationally expensive for large-scale problems. Sparsity is employed to handle noise in the data and to promote interpretability of a learning model. Although the use of a convex ℓ 1 regularizer to encourage sparsity is common practice, the combinatorial ℓ 0 penalty often has more favorable statistical properties. In this paper, we directly constrain sparsity by specifying a maximally allowable number of nonzeros, in other words, by imposing an ℓ 0 constraint. Here, we introduce an efficient approximate Newton algorithm using warm starts for solving the nonconvex ℓ 0 -constrained inverse covariance learning problem. Numerical experiments on standard data sets show that the performance of the proposed algorithm is competitive with state-of-the-art methods.

$\ell_0$-Constrained↗

DOE OSTI2018

Stochastic Design Optimization of Microstructural Features Using Linear Programming for Robust Design

Microstructure design can have a substantial effect on the performance of critical components in numerous aerospace applications. However, the stochastic nature of metallic microstructures leads to deviations in material properties from the design point and alters the performance of these critical components. In this paper, a novel stochastic linear programming (LP) methodology is developed for microstructure design accounting for uncertainties in desired properties. The metallic microstructure is represented using a finite element discretized form of the orientation distribution function (ODF). The inverse LP problem solves the mean values and covariance matrix of the ODFs to maximize the mean values of a property, given the statistical constraints on other properties. The highlight is an analytical uncertainty quantification model via a Gaussian distribution to model propagation of microstructural uncertainties to the properties. Examples illustrate maximization of the yield strength and magnetostriction of a galfenol alloy when constrained by uncertainties in a set of stiffness constants.

42 ENGINEERING↗

DOE OSTI2020

Adaptive Activation Functions Accelerate Convergence in Deep and Physics-informed Neural Networks

We employ adaptive activation functions for regression in deep and physics-informed neural networks (PINNs) to approximate smooth and discontinuous functions as well as solutions of linear and nonlinear partial differential equations. In particular, we solve the nonlinear Klein-Gordon equation, which has smooth solutions, the nonlinear Burgers equation, which can admit high gradient solutions, and the Helmholtz equation. We introduce a scalable hyper-parameter in the activation function, which can be optimized to achieve best performance of the network as it changes dynamically the topology of the loss function involved in the optimization process. The adaptive activation function has better learning capabilities than the traditional one (fixed activation) as it improves greatly the convergence rate, especially at early training, as well as the solution accuracy. To better understand the learning process, we plot the neural network solution in the frequency domain to examine how the network captures successively different frequency bands present in the solution. We consider both forward problems, where the approximate solutions are obtained, as well as inverse problems, where parameters involved in the governing equation are identified. Our simulation results show that the proposed method is a very simple and effective approach to increase the efficiency, robustness and accuracy of the neural network approximation of nonlinear functions as well as solutions of partial differential equations, especially for forward problems. We theoretically prove that in the proposed method, gradient descent algorithms are not attracted to suboptimal critical points or local minima.

machine leaning, Bad minima, Inverse problems, Phy↗