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DOE OSTI2022

Discovering causal structure with reproducing-kernel Hilbert space ε -machines

We merge computational mechanics’ definition of causal states (predictively equivalent histories) with reproducing-kernel Hilbert space (RKHS) representation inference. The result is a widely applicable method that infers causal structure directly from observations of a system’s behaviors whether they are over discrete or continuous events or time. A structural representation—a finite- or infinite-state kernel ϵ-machine—is extracted by a reduced-dimension transform that gives an efficient representation of causal states and their topology. In this way, the system dynamics are represented by a stochastic (ordinary or partial) differential equation that acts on causal states. We introduce an algorithm to estimate the associated evolution operator. Paralleling the Fokker–Planck equation, it efficiently evolves causal-state distributions and makes predictions in the original data space via an RKHS functional mapping. We demonstrate these techniques, together with their predictive abilities, on discrete-time, discrete-value infinite Markov-order processes generated by finite-state hidden Markov models with (i) finite or (ii) uncountably infinite causal states and (iii) continuous-time, continuous-value processes generated by thermally driven chaotic flows. The method robustly estimates causal structure in the presence of varying external and measurement noise levels and for very high-dimensional data.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1990

Least-squares finite element methods for compressible Euler equations

A method based on backward finite differencing in time and a least-squares finite element scheme for first-order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the L-sq-norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high-order elements, computed solutions based on the L-sq method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted H1-norm of the residual is proposed and leads to a successful scheme with high-degree elements. Finally, a conservative least-squares finite element method is also developed. Numerical results for two-dimensional problems are given to demonstrate the shock resolution of the methods and compare different approaches.

Jiang, Bo-Nan↗

DOE OSTI2022

Operator inference for non-intrusive model reduction with quadratic manifolds

This paper proposes a novel approach for learning a data-driven quadratic manifold from high-dimensional data, then employing this quadratic manifold to derive efficient physics-based reduced-order models. The key ingredient of the approach is a polynomial mapping between high-dimensional states and a low-dimensional embedding. This mapping consists of two parts: a representation in a linear subspace (computed in this work using the proper orthogonal decomposition) and a quadratic component. The approach can be viewed as a form of data-driven closure modeling, since the quadratic component introduces directions into the approximation that lie in the orthogonal complement of the linear subspace, but without introducing any additional degrees of freedom to the low-dimensional representation. Combining the quadratic manifold approximation with the operator inference method for projection-based model reduction leads to a scalable non-intrusive approach for learning reduced-order models of dynamical systems. Further, applying the new approach to transport-dominated systems of partial differential equations illustrates the gains in efficiency that can be achieved over approximation in a linear subspace.

42 ENGINEERING↗

DOE OSTI2020

MEUMAPPS (Microstructure Evolution Using Massively Parallel Phase-field Simulations)

The software “MEUMAPPS” is a h high-performance computing code used to simulate microstructure evolution associated with diffusional solid-state transformations in structural alloys. Understanding microstructure evolution during thermo-mechanical processing of structural alloys is the first step towards designing and processing alloys for specific applications by meeting property requirements demanded by the application. In this instance, the code is an important component in predicting processing-structure linkages during additive manufacturing of structural alloys as a function of processing parameters and alloy composition used in an additive manufacturing process. The code provides a detailed three-dimensional distribution of different types of phases / constituents and their morphologies, as well as the compositions of these phases that make up the microstructure. The microstructure is simulated by solving the governing partial differential equations using a Fourier Spectral Method.

Radhakrishnan, Balasubram↗

DOE OSTI2023

Embedding hard physical constraints in neural network coarse-graining of three-dimensional turbulence

In recent years, deep learning approaches have shown much promise in modeling complex systems in the physical sciences. A major challenge in deep learning of partial differential equations is enforcing physical constraints and boundary conditions. In this work, we propose a general framework to directly embed the notion of an incompressible fluid into convolutional neural networks, and apply this to coarse-graining of turbulent flow. These physics-embedded neural networks leverage interpretable strategies from numerical methods and computational fluid dynamics to enforce physical laws and boundary conditions by taking advantage the mathematical properties of the underlying equations. Here, we demonstrate results on three-dimensional fully developed turbulence, showing that this technique drastically improves local conservation of mass, without sacrificing performance according to several other metrics characterizing the fluid flow.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Tackling the curse of dimensionality in fractional and tempered fractional PDEs with physics-informed neural networks

Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are mesh-based, thus struggling with the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a promising solution due to their universal approximation, generalization ability, and mesh-free training. In principle, Monte Carlo fractional PINN (MC-fPINN) estimates fractional derivatives using Monte Carlo methods and thus could lift CoD. However, this may cause significant variance and errors, hence affecting convergence; in addition, MC-fPINN is sensitive to hyperparameters. In general, numerical methods and specifically PINNs for tempered fractional PDEs are under-developed. Herein, we extend MC-fPINN to tempered fractional PDEs to address these issues, resulting in the Monte Carlo tempered fractional PINN (MC-tfPINN). To reduce possible high variance and errors from Monte Carlo sampling, we replace the one-dimensional (1D) Monte Carlo with 1D Gaussian quadrature, applicable to both MC-fPINN and MC-tfPINN. We validate our methods on various forward and inverse problems of fractional and tempered fractional PDEs, scaling up to 100,000 dimensions. Our improved MC-fPINN/MC-tfPINN using quadrature consistently outperforms the original versions in accuracy and convergence speed in very high dimensions.

42 ENGINEERING↗

DOE OSTI2024

Accurate data-driven surrogates of dynamical systems for forward propagation of uncertainty

Stochastic collocation (SC) is a well-known non-intrusive method of constructing surrogate models for uncertainty quantification. In dynamical systems, SC is especially suited for full-field uncertainty propagation that characterizes the distributions of the high-dimensional solution fields of a model with stochastic input parameters. However, due to the highly nonlinear nature of the parameter-to-solution map in even the simplest dynamical systems, the constructed SC surrogates are often inaccurate. Here, this work presents an alternative approach, where we apply the SC approximation over the dynamics of the model, rather than the solution. By combining the data-driven sparse identification of nonlinear dynamics framework with SC, we construct dynamics surrogates and integrate them through time to construct the surrogate solutions. We demonstrate that the SC-over-dynamics framework leads to smaller errors, both in terms of the approximated system trajectories as well as the model state distributions, when compared against full-field SC applied to the solutions directly. We present numerical evidence of this improvement using three test problems: a chaotic ordinary differential equation, and two partial differential equations from solid mechanics.

42 ENGINEERING↗

DOE OSTI2026

A Tensor Network-Based Quantum Algorithm for the Nonlinear 1D Burgers' Equation

In this work, we implement a tensor network-based quantum algorithm to solve unsteady, nonlinear partial differential equations (PDEs). The challenge lies in how to effectively represent, encode, process, and evolve the nonlinear system of PDEs on quantum computers. We will discuss the new techniques using the compressible 1-dimensional (1D) Burgers' equation as an example, because it represents the fundamental nonlinear feature and yet removes certain complexity in physics, allowing us to focus on the design of quantum algorithms. Previous attempts to solve nonlinear PDEs in quantum computation have often involved storing multiple copies of solutions or employing linearizations. Neither is practical due to exponential scaling with evolution time or insufficient solution accuracy. Our framework is based on matrix product states (MPSs) and matrix product operators (MPOs). For example, the velocity field is represented by MPS, whereas the linear and nonlinear spatial differential terms of the velocity field are processed by MPOs. Our primary focus herein is to verify and validate the various tensor network components of the algorithm using solutions obtained by the classical algorithms on high performance computing (HPC) architectures. We use a classical time marching method to demonstrate the functionality of the tensor network operations to model the PDE and their robustness with the time evolution of the system. Our classical simulation results demonstrate the utility of tensor network-based operations in modeling nonlinear PDEs and highlight the necessity as well as potential advantages of using quantum simulations for these techniques.

Gopalakrishnan Meena, Murali [ORNL] (ORCID:0000000↗

NASA NTRS1994

Theoretical study of the incompressible Navier-Stokes equations by the least-squares method

Usually the theoretical analysis of the Navier-Stokes equations is conducted via the Galerkin method which leads to difficult saddle-point problems. This paper demonstrates that the least-squares method is a useful alternative tool for the theoretical study of partial differential equations since it leads to minimization problems which can often be treated by an elementary technique. The principal part of the Navier-Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems, so the three-dimensional div-curl system is thoroughly studied at first. By introducing a dummy variable and by using the least-squares method, this paper shows that the div-curl system is properly determined and elliptic, and has a unique solution. The same technique then is employed to prove that the Stokes equations are properly determined and elliptic, and that four boundary conditions on a fixed boundary are required for three-dimensional problems. This paper also shows that under four combinations of non-standard boundary conditions the solution of the Stokes equations is unique. This paper emphasizes the application of the least-squares method and the div-curl method to derive a high-order version of differential equations and additional boundary conditions. In this paper, an elementary method (integration by parts) is used to prove Friedrichs' inequalities related to the div and curl operators which play an essential role in the analysis.

Jiang, Bo-Nan↗

DOE OSTI2022

Quantum-inspired method for solving the Vlasov-Poisson equations

Kinetic simulations of collisionless (or weakly collisional) plasmas using the Vlasov equation are often infeasible due to high-resolution requirements and the exponential scaling of computational cost with respect to dimension. Recently, it has been proposed that matrix product state (MPS) methods, a quantum-inspired but classical algorithm, can be used to solve partial differential equations with exponential speed-up, provided that the solution can be compressed and efficiently represented as a MPS within some tolerable error threshold. Here, in this work, we explore the practicality of MPS methods for solving the Vlasov-Poisson equations for systems with one coordinate in space and one coordinate in velocity, and find that important features of linear and nonlinear dynamics, such as damping or growth rates and saturation amplitudes, can be captured while compressing the solution significantly. Furthermore, by comparing the performance of different mappings of the distribution functions onto the MPS, we develop an intuition of the MPS representation and its behavior in the context of solving the Vlasov-Poisson equations, which will be useful for extending these methods to higher-dimensional problems.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Multi-fidelity Bayesian neural networks: Algorithms and applications

Here we propose a new class of Bayesian neural networks (BNNs) that can be trained using noisy data of variable fidelity, and we apply them to learn function approximations as well as to solve inverse problems based on partial differential equations (PDEs). These multi-fidelity BNNs consist of three neural networks: The first is a fully connected neural network, which is trained following the maximum a posteriori probability (MAP) method to fit the low-fidelity data; the second is a Bayesian neural network employed to capture the cross-correlation with uncertainty quantification between the low- and high-fidelity data; and the last one is the physics-informed neural network, which encodes the physical laws described by PDEs. For the training of the last two neural networks, we first employ the mean-field variational inference (VI) to maximize the evidence lower bound (ELBO) to obtain informative prior distributions for the hyperparameters in the BNNs, and subsequently we use the Hamiltonian Monte Carlo (HMC) method to estimate accurately the posterior distributions for the corresponding hyperparameters. We demonstrate the accuracy of the present method using synthetic data as well as real measurements. Specifically, we first approximate a one- and four-dimensional function, and then infer the reaction rates in one- and two-dimensional diffusion-reaction systems. Moreover, we infer the sea surface temperature (SST) in the Massachusetts and Cape Cod Bays using satellite images and in-situ measurements. Taken together, our results demonstrate that the present method can capture both linear and nonlinear correlation between the low- and high-fidelity data adaptively, identify unknown parameters in PDEs, and quantify uncertainties in predictions, given a few scattered noisy high-fidelity data. Finally, we demonstrate that we can effectively and efficiently reduce the uncertainties and hence enhance the prediction accuracy with an active learning approach, using as examples a specific one-dimensional function approximation and an inverse PDE problem.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Online randomized interpolative decomposition with a posteriori error estimator for temporal PDE data reduction

Traditional low-rank approximation is a powerful tool for compressing large data matrices that arise in simulations of partial differential equations (PDEs), but suffers from high computational cost and requires several passes over the PDE data. The compressed data may also lack interpretability thus making it difficult to identify feature patterns from the original data. Here, to address these issues, we present an online randomized algorithm to compute the interpolative decomposition (ID) of large-scale data matrices in situ. Compared to previous randomized IDs that used the QR decomposition to determine the column basis, we adopt a streaming ridge leverage score-based column subset selection algorithm that dynamically selects proper basis columns from the data and thus avoids an extra pass over the data to compute the coefficient matrix of the ID. In particular, we adopt a single-pass error estimator based on the non-adaptive Hutch++ algorithm to provide real-time error approximation for determining the best coefficients. As a result, our approach only needs a single pass over the original data and thus is suitable for large and high-dimensional matrices stored outside of core memory or generated in PDE simulations. A strategy to improve the accuracy of the reconstructed data gradient, when desired, within the ID framework is also presented. We provide numerical experiments on turbulent channel flow and ignition simulations, and on the NSTX Gas Puff Image dataset, comparing our algorithm with the offline ID algorithm to demonstrate its utility in real-world applications.

Column subset selection↗

NASA NTRS2020

On Boundary-Value Problems for RANS Equations and Two-Equation Turbulence Models

Currently, in engineering computations for high Reynolds number turbulent flows, turbulence modeling continues to be the most frequently used approach to represent the effects of turbulence. Such models generally rely on solving either one or two transport equations along with the Reynolds-Averaged Navier–Stokes (RANS) equations. The solution of the boundary-value problem of any system of partial differential equations requires the complete delineation of the equations and the boundary conditions, including any special restrictions and conditions. In the literature, such a description is often incomplete, neglecting important details related to the boundary conditions and possible restrictive conditions, such as how to ensure satisfying prescribed values of the dependent variables of the transport equations in the far field of a finite domain. In this article, we discuss the possible influence of boundary values, as well as near-field and far-field behavior, on the solution of the RANS equations coupled with transport equations for turbulence modeling. In so doing, we defne the concept of a welldefined boundary-value problem. Additionally, a three-dimensional, rather than a simpler one-dimensional analysis is performed to analyze the near-wall and far-field behavior of the turbulence model variables. This allows an assessment of the decay rate of these variables required to realize the boundary conditions in the far field. This paper also addresses the impact of various transformations of two-equation models (e.g., the model of Wilcox) to remove the singular behavior of the dissipation rate (ω) at the surface boundary. Finally, the issue of well-posedness regarding the governing equations is considered. A compelling argument (although not a proof) for ill-posedness is made for both direct and inverse problems.

Navier–Stokes equations↗

DOE OSTI2020

Transported PDF Modeling of Compressible Turbulent Reactive Flows by using the Eulerian Monte Carlo Fields Method

Although the transported probability density function (PDF) method has been developed for decades, its application has been mainly focused on the low-Mach number flow problems. This work extends the transported PDF method to compressible flow problems. The Eulerian Monte Carlo fields (EMCF) solution method is employed to solve the transported PDF equation for compressible flow problems. A pseudo stagnation enthalpy is introduced and its stochastic partial differential equation is derived to ensure total energy conservation numerically. A new mixing model called interaction by partial exchange with mean (IPEM) is introduced to expand the available choices of mixing models for the EMCF method. The consistency of the EMCF method is examined for solving the transported PDF equation. Numerical implementation details are discussed, such as the density coupling between the compressible flow solver and the EMCF solver, discretization schemes for the mixing terms and the stochastic terms. The implemented compressible flow solver coupled with the EMCF solver is verified and validated in a series of test cases with increasing level of complexity, ranging from a statistically one-dimensional turbulent mixing layer to a self-excited resonance model rocket combustor. It is observed that in general with the increase of compressibility, there is an increase in the sensitivity of the modeling results to the different models and algorithms. This makes it necessary to develop a thorough understanding of the model sensitivity in order to develop a robust and accurate simulation solver for highly compressible turbulent reactive flows. The thermo-acoustic instability inside the model rocket combustor case is captured reasonably, which demonstrates the overall capability of the developed compressible turbulent combustion solver based on the transported PDF method.

42 ENGINEERING↗

DOE OSTI2019

A space-time adaptive finite element method with exponential time integrator for the phase field model of pitting corrosion

In this paper we propose a space-time adaptive finite element method for the phase field model of pitting corrosion, which is a parabolic partial differential equation system consisting of a phase variable and a concentration variable. A major challenge in solving this phase field model is that the problem is very stiff, which makes the time step size extremely small for standard temporal discretizations. Another difficulty is that a high spatial resolution is required to capture the steep gradients within the diffused interface, which results in very large number of degrees of freedom for uniform meshes. To overcome the stiffness of this model, we combine the Rosenbrock–Euler exponential integrator with Crank–Nicolson scheme for the temporal discretization. Moreover, by exploiting the fact that the speed of the corroding interface decreases with time, we derive an adaptive time stepping formula. For the spatial approximation, we propose a simple and efficient strategy to generate adaptive meshes that reduces the computational cost significantly. Thus, the proposed method utilizes local adaptivity and mesh refinement for efficient simulation of the corrosive dissolution over long times in heterogeneous media with complex microstructures. We also present an extensive set of numerical experiments in both two and three dimensional spaces to demonstrate efficiency and robustness of the proposed method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Predicting rare events using neural networks and short-trajectory data

Estimating the likelihood, timing, and nature of events is a major goal of modeling stochastic dynamical systems. When the event is rare in comparison with the timescales of simulation and/or measurement needed to resolve the elemental dynamics, accurate prediction from direct observations becomes challenging. In such cases a more effective approach is to cast statistics of interest as solutions to Feynman-Kac equations (partial differential equations). Here, we develop an approach to solve Feynman-Kac equations by training neural networks on short-trajectory data. Our approach is based on a Markov approximation but otherwise avoids assumptions about the underlying model and dynamics. This makes it applicable to treating complex computational models and observational data. Additionally, we illustrate the advantages of our method using a low-dimensional model that facilitates visualization, and this analysis motivates an adaptive sampling strategy that allows on-the-fly identification of and addition of data to regions important for predicting the statistics of interest. Finally, we demonstrate that we can compute accurate statistics for a 75-dimensional model of sudden stratospheric warming. This system provides a stringent test bed for our method.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

SPIKANs: separable physics-informed Kolmogorov–Arnold networks

Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) in scientific computing. While PINNs typically use multilayer perceptrons (MLPs) as their underlying architecture, recent advancements have explored alternative neural network structures. One such innovation is the Kolmogorov–Arnold Network (KAN), which has demonstrated benefits over traditional MLPs, including faster neural scaling and better interpretability. The application of KANs to physics-informed learning has led to the development of Physics-Informed KANs (PIKANs), enabling the use of KANs to solve PDEs. However, despite their advantages, KANs often suffer from slower training speeds, particularly in higher-dimensional problems where the number of collocation points grows exponentially with the dimensionality of the system. To address this challenge, we introduce Separable Physics-Informed Kolmogorov–Arnold Networks (SPIKANs). This novel architecture applies the principle of separation of variables to PIKANs, decomposing the problem such that each dimension is handled by an individual KAN. This approach drastically reduces the computational complexity of training without sacrificing accuracy, facilitating their application to higher-dimensional PDEs. Through a series of benchmark problems, we demonstrate the effectiveness of SPIKANs, showcasing their superior scalability and performance compared to PIKANs and highlighting their potential for solving complex, high-dimensional PDEs in scientific computing.

Kolmogorov-Arnold networks↗

DOE OSTI2023

Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems

We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability to approximate the parameter-to-solution maps defined by PDEs using as training data solutions of PDEs at a limited number of parameter samples. The computational cost of BIPs can be drastically reduced if the large number of PDE solves required for posterior characterization are replaced with evaluations of trained neural operators. However, reducing error in the resulting BIP solutions via reducing the approximation error of the neural operators in training can be challenging and unreliable. We provide an a priori error bound result that implies certain BIPs can be ill-conditioned to the approximation error of neural operators, thus leading to inaccessible accuracy requirements in training. To reliably deploy neural operators in BIPs, we consider a strategy for enhancing the performance of neural operators: correcting the prediction of a trained neural operator by solving a linear variational problem based on the PDE residual. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error, all while retaining substantial computational speedups of posterior sampling when models are governed by highly nonlinear PDEs. The strategy is applied to two numerical examples of BIPs based on a nonlinear reaction–diffusion problem and deformation of hyperelastic materials. We demonstrate that posterior representations of the two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction.

97 MATHEMATICS AND COMPUTING↗