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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
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 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↗
DOE OSTI2026

Causality-respecting adaptive refinement for PINNs: enabling precise interface evolution in phase field modeling

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving physical systems described by partial differential equations (PDEs). However, their accuracy in dynamical systems, particularly those involving sharp moving boundaries with complex initial morphologies, remains a challenge. Here, this study introduces an approach combining residual-based adaptive refinement (RBAR) with causality-informed training to enhance the performance of PINNs in solving spatio-temporal PDEs. Our method employs a three-step iterative process: initial causality-based training, RBAR-guided domain refinement, and subsequent causality training on the refined mesh. Applied to the Allen-Cahn equation, a widely-used model in phase field simulations, our approach demonstrates significant improvements in solution accuracy and computational efficiency over traditional PINNs. Notably, we observe an ‘overshoot and relocate’ phenomenon in dynamic cases with complex morphologies, showcasing the method’s adaptive error correction capabilities. This synergistic interaction between RBAR and causality training enables accurate capture of interface evolution, even in challenging scenarios where traditional PINNs fail. Our framework not only resolves the limitations of uniform refinement strategies but also provides a generalizable methodology for solving a broad range of spatio-temporal PDEs. The enhanced performance of the RBAR–causality combined framework demonstrates its strong potential for advancing PINN-based modeling of physical systems characterized by complex, evolving interfaces.

Allen-Cahn equations↗
DOE OSTI2026

Higher-order LaSDI: Reduced order modeling with multiple time derivatives

Solving complex partial differential equations (PDEs) is essential across scientific disciplines but often requires numerical models that can be prohibitively expensive in time-sensitive applications. Reduced-order models (ROMs) address this challenge by exploiting low-dimensional structure to create fast approximations. The Latent Space Dynamics Identification (LaSDI) framework has demonstrated success in learning ROMs for parameterized PDE families, but remains limited to first-order systems. Here, in this paper, we propose Higher-Order LaSDI (HLaSDI), which extends the LaSDI framework to PDEs with arbitrary order of time derivatives. This generalization significantly expands the applicability of LaSDI-based methods to systems previously outside their scope, including hyperbolic PDEs. We demonstrate HLaSDI’s accuracy and efficiency on several linear and nonlinear benchmark problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Solving sparse finite element problems on neuromorphic hardware

The finite element method (FEM) is one of the most important and ubiquitous numerical methods for solving partial differential equations (PDEs) on computers for scientific and engineering discovery. Applying the FEM to larger and more detailed scientific models has driven advances in high-performance computing for decades. Here we demonstrate that scalable spiking neuromorphic hardware can directly implement the FEM by constructing a spiking neural network that solves the large, sparse, linear systems of equations at the core of the FEM. We show that for the Poisson equation, a fundamental PDE in science and engineering, our neural circuit achieves meaningful levels of numerical accuracy and close to ideal scaling on modern, inherently parallel and energy-efficient neuromorphic hardware, specifically Intel’s Loihi 2 neuromorphic platform. We illustrate extensions to irregular mesh geometries in both two and three dimensions as well as other PDEs such as linear elasticity. Our spiking neural network is constructed from a recurrent network model of the brain’s motor cortex and, in contrast to black-box deep artificial neural network-based methods for PDEs, directly translates the well-understood and trusted mathematics of the FEM to a natively spiking neuromorphic algorithm.

Applied mathematics↗
DOE OSTI2025

Towards a Quantum Algorithm for the Incompressible Nonlinear Navier-Stokes Equations

In this work, we present novel concepts for quantum algorithms to solve transient, 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 incompressible Navier-Stokes equations 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 solving 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. We propose a new framework based on matrix product states (MPSs) and matrix product operators (MPOs), in addition to the Krylov subspace methods. For example, the solution variables of the Navier-Stokes equations are represented by MPSs, and the linear and nonlinear terms are processed by MPOs. The time evolution of the operators is attained by a fast-forwarding algorithm using Krylov subspace methods. Furthermore, we discuss various techniques for efficient encoding of MPSs, measurement reduction for MPOs, and use of tensor operations to treat multi-variate, multi-physics characteristics of Navier-Stokes.

Gopalakrishnan Meena, Murali [ORNL] (ORCID:0000000↗
DOE OSTI2026

Dual Representations and H ∞ -Optimal Control of Partial Differential Equations

We consider H ∞ -optimal state-feedback control of the class of linear Partial Differential Equations (PDEs) which admit a Partial Integral Equation (PIE) representation. While linear matrix inequalities are commonly used for optimal control of Ordinary Differential Equations (ODEs), the absence of a universal state-space representation and suitable dual form prevents such methods from being applied to optimal control of PDEs. Specifically, for ODEs, the controller synthesis problem is defined in state-space, and duality is used to resolve the bilinearity of that synthesis problem. Recently, the PIE representation was proposed as a universal state-space representation for linear PDE systems. In this paper, we show that any PDE system represented by a PIE admits a dual PIE with identical stability and I/O properties. This result allows us to reformulate the stabilizing and optimal state-feedback control problems as convex optimization over the cone of positive Partial Integral (PI) operators. Operator inversion formulae then allow us to construct feedback gains for the original PDE system. The results are verified through application to several canonical problems in optimal control of PDEs and indicate the resulting bounds on H ∞ norm are not conservative.

42 ENGINEERING↗
DOE OSTI2026

Explicit Monotone Stable Super-Time-stepping Methods for Finite Time Singularities

We explore a novel way to numerically resolve the scaling behavior of finite-time singularities in solutions of nonlinear parabolic PDEs. The Runge–Kutta–Legendre (RKL) and Runge–Kutta–Gegenbauer (RKG) super-time-stepping methods were originally developed for nonlinear complex physics problems with diffusion. These are multistage single step second-order, forward-in-time methods with no implicit solves. The advantage is that the time-step size for stability scales with stage number 𝑠 as $\mathcal{O}$⁡(𝑠 2 ). Many interesting nonlinear PDEs have finite-time singularities, and the presence of diffusion often limits one to using implicit or semi-implicit time-step methods for stability constraints. Finite-time singularities are particularly challenging due to the large range of scales that one desires to resolve, often with adaptive spatial grids and adaptive time steps. Here, in this study, we show two examples of nonlinear PDEs for which the self-similar singularity structure has time and space scales that are resolvable using the RKL and RKG methods, without forcing even smaller time steps. Compared to commonly used implicit numerical methods, we achieve a significantly smaller run time while maintaining comparable accuracy. We also prove numerical monotonicity for both the RKL and RKG methods under their linear stability conditions for the constant coefficient heat equation, in the case of infinite domain and periodic boundary condition, leading to a theoretical guarantee of the superiority of the RKL and RKG methods over traditional super-time-stepping methods, such as the Runge-Kutta-Chebyshev and the orthogonal Runge-Kutta-Chebyshev methods. Code can be found at https://github.com/ZT220501/SRK-Singularity.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Machine-learning based model reduction for partial differential equations

We develop a novel synergistic approach between model reduction and machine learning. The specific goal of this project is to aid in the construction of reduced order models for basis functions that are custom-made to represent the solution of partial differential equations. Partial differential equations (PDEs) are one of the main mathematical tools for describing physical phenomena. However, due to either efficiency or necessity, for many real-world problems, we are interested in constructing reduced order models (ROMs) which focus only on the explicit computation of subsets of the active spatio-temporal scales in the problem, while treating the interaction with the rest of the scales approximately. The task of accurate representation of such interactions (usually called memory terms) constitutes a vast area of research known as model reduction. PI Stinis has significant expertise in the construction of ROMs for complex systems. In addition, in recent work with the project key participant Qadeer, they have utilized machine learning to acquire custom-made basis functions (CBFs) to expand the solutions of PDEs. In the proposed work, we will merge the two concepts by constructing ROMs for subsets of the CBFs needed to represent the solution of a PDE. Specifically, we will use the Mori-Zwanzig model reduction formalism to construct ROMs for subsets of CBFs for nonlinear PDEs of various complexity, as well as investigate the usage of CBFs in the spectral vanishing viscosity method for problems that can form shocks in finite time. The outcome of the research is aimed to be proof-of-concept about a novel synergistic approach between model reduction and machine learning, thus advancing the field of scientific machine learning. Such a capability will benefit the efficient modeling of physical systems appearing in various areas of interest to the DOE.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Latent Twins

Over the past decade, scientific machine learning has transformed the development of mathematical and computational frameworks for analyzing, modeling, and predicting complex systems. From inverse problems to numerical partial differential equations (PDEs), dynamical systems, and model reduction, these advances have pushed the boundaries of what can be simulated. Yet they have often progressed in parallel, with representation learning and algorithmic solution methods evolving largely as separate pipelines. With Latent Twins, we propose a unifying mathematical framework that creates a hidden surrogate in latent space for the underlying equations. Whereas digital twins mirror physical systems in the digital world, Latent Twins mirror mathematical systems in a learned latent space governed by operators. Through this lens, classical modeling, inversion, model reduction, and operator approximation all emerge as special cases of a single principle. We establish the fundamental approximation properties of Latent Twins for both ordinary differential equations (ODEs) and PDEs and demonstrate the framework across three representative settings: (i) canonical ODEs, capturing diverse dynamical regimes; (ii) a PDE benchmark using the shallow-water equations, contrasting Latent Twin simulations with deep operator network and forecasts with a four-dimensional variational method baseline; and (iii) a challenging real-data geopotential reanalysis dataset, reconstructing and forecasting from sparse, noisy observations. Latent Twins provide a compact, interpretable surrogate for solution operators that evaluate across arbitrary time gaps in a single-shot, while remaining compatible with scientific pipelines such as assimilation, control, and uncertainty quantification. Looking forward, this framework offers scalable, theory-grounded surrogates that bridge data-driven representation learning and classical scientific modeling across disciplines.

Latent Twins↗
DOE OSTI2026

Multiscale Neural Networks for Approximating Green’s Functions

Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Green’s functions. However, Green’s functions are notoriously difficult to learn due to their poor regularity, which typically requires larger NNs and longer training times. In this work, we address these challenges by leveraging multiscale NNs to learn Green’s functions. Through theoretical analysis using multiscale Barron space methods and experimental validation, we show that the multiscale approach significantly reduces the necessary NN size and accelerates training.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Boundary Corrections for Kernel Approximation to Differential Operators

The kernel-based approach to operator approximation for partial differential equations has been shown to be unconditionally stable for linear PDEs and numerically exhibit unconditional stability for non-linear PDEs. These methods have the same computational cost as an explicit finite difference scheme but can exhibit order reduction at boundaries. In previous work on periodic domains, order reduction was addressed, yielding high-order accuracy. The issue addressed in this work is the elimination of order reduction of the kernel-based approach for a more general set of boundary conditions. Further, we consider the case of both first and second order operators. To demonstrate the theory, we provide not only the mathematical proofs but also experimental results by applying various boundary conditions to different types of equations. The results agree with the theory, demonstrating a systematic path to high order for kernel-based methods on bounded domains.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A review of high order strong stability preserving two-derivative explicit, implicit, and IMEX methods

High order strong stability preserving time discretizations ensure the nonlinear non-inner-product strong stability properties of spatial discretizations suited for the stable simulation of hyperbolic PDEs in a wide variety of application areas including fluid dynamics, magnetohydrodynamics, semiconductor devices, electromagnetics, and astrophysics. Over the past decade multiderivative time-stepping have been increasingly used for the time-evolution hyperbolic PDEs, so that the strong stability properties of these methods have become important. In this work we review sufficient conditions for a two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and different conditions on the second derivative. In particular we present the strong stability preserving theory for explicit and implicit two-derivative Runge–Kutta schemes, including a special condition on the second derivative under which these implicit methods may be unconditionally strong stability preserving. This special condition is natural for the stiff component of wide range of plasma physics problems, and can be useful in the context of strong stability preserving implicit-explicit multi-derivative Runge–Kutta schemes, where the time-step restriction is then independent of the stiff term. Lastly, we present the strong stability preserving theory for implicit-explicit multi-derivative general linear methods, and some novel second and third order methods where the time-step restriction is independent of the stiff term.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Exact enforcement of temporal continuity in sequential physics-informed neural networks

The use of deep learning methods in scientific computing represents a potential paradigm shift in engineering problem solving. One of the most prominent developments is Physics-Informed Neural Networks (PINNs), in which neural networks are trained to satisfy partial differential equations (PDEs). While this method shows promise, the standard version has been shown to struggle in accurately predicting the dynamic behavior of time-dependent problems. To address this challenge, methods have been proposed that decompose the time domain into multiple segments, employing a distinct neural network in each segment and directly incorporating continuity between them in the loss function of the minimization problem. In this work we introduce a method to exactly enforce continuity between successive time segments via a solution ansatz. This hard constrained sequential PINN (HCS-PINN) method is simple to implement and eliminates the need for any loss terms associated with temporal continuity. The method is tested for a number of benchmark problems involving both linear and non-linear PDEs. Examples include various first order time dependent problems in which traditional PINNs struggle, namely advection, Allen–Cahn, and Korteweg–de Vries equations. Furthermore, second and third order time-dependent problems are demonstrated via wave and Jerky dynamics examples, respectively. Notably, the Jerky dynamics problem is chaotic, making the problem especially sensitive to temporal accuracy. Finally, the numerical experiments conducted with the proposed method demonstrated superior convergence and accuracy over both traditional PINNs and the soft-constrained counterparts.

42 ENGINEERING↗
DOE OSTI2025

Constrained or unconstrained? Neural-network-based equation discovery from data

Throughout many fields, practitioners often rely on differential equations to model systems. Yet, for many applications, the theoretical derivation of such equations and/or the accurate resolution of their solutions may be intractable. Instead, recently developed methods, including those based on parameter estimation, operator subset selection, and neural networks, allow for the data-driven discovery of both ordinary and partial differential equations (PDEs), on a spectrum of interpretability. The success of these strategies is often contingent upon the correct identification of representative equations from noisy observations of state variables and, as importantly and intertwined with that, the mathematical strategies utilized to enforce those equations. Specifically, the latter has been commonly addressed via unconstrained optimization strategies. Representing the PDE as a neural network, we propose to discover the PDE (or the associated operator) by solving a constrained optimization problem and using an intermediate state representation similar to a physics-informed neural network (PINN). The objective function of this constrained optimization problem promotes matching the data, while the constraints require that the discovered PDE is satisfied at a number of spatial collocation points. We present a penalty method and a widely used trust-region barrier method to solve this constrained optimization problem, and we compare these methods on numerical examples. Our results on several example problems demonstrate that the latter constrained method outperforms the penalty method, particularly for higher noise levels or fewer collocation points. This work motivates further exploration into using sophisticated constrained optimization methods in scientific machine learning, as opposed to their commonly used, penalty-method or unconstrained counterparts. For both of these methods, we solve these discovered neural network PDEs with classical methods, such as finite difference methods, as opposed to PINNs-type methods relying on automatic differentiation. Here, we briefly highlight how simultaneously fitting the data while discovering the PDE improves the robustness to noise and other small, yet crucial, implementation details.

Data-driven discovery↗
DOE OSTI2025

Optimizing the optimizer for physics-informed neural networks and Kolmogorov-Arnold networks

Physics-Informed Neural Networks (PINNs) have revolutionized the computation of PDE solutions by integrating partial differential equations (PDEs) into the neural network’s training process as soft constraints, becoming an important component of the scientific machine learning (SciML) ecosystem. More recently, physics-informed Kolmogorv-Arnold networks (PIKANs) have also shown to be effective and comparable in accuracy with PINNs. In their current implementation, both PINNs and PIKANs are mainly optimized using first-order methods like Adam, as well as quasi-Newton methods such as BFGS and its low-memory variant, L-BFGS. However, these optimizers often struggle with highly nonlinear and non-convex loss landscapes, leading to challenges such as slow convergence, local minima entrapment, and (non)degenerate saddle points. In this study, we investigate the performance of Self- Scaled BFGS (SSBFGS), Self-Scaled Broyden (SSBroyden) methods and other advanced quasi-Newton schemes, including BFGS and L-BFGS with different line search strategies. These methods dynamically rescale updates based on historical gradient information, thus enhancing training efficiency and accuracy. We systematically compare these optimizers – using both PINNs and PIKANs – on key challenging PDEs, including the Burgers, Allen-Cahn, Kuramoto-Sivashinsky, Ginzburg-Landau, and Stokes equations. Additionally, we evaluate the performance of SSBFGS and SSBroyden for Deep Operator Network (DeepONet) architectures, demonstrating their effectiveness for data-driven operator learning. Our findings provide state-of-the-art results with orders-of-magnitude accuracy improvements without the use of adaptive weights or any other enhancements typically employed in PINNs. More broadly, our work reveal insights into the effectiveness of quasi-Newton optimization strategies in significantly improving the convergence and accurate generalization of PINNs and PIKANs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A higher-order finite-element implementation of the nonlinear Fokker–Planck collision operator for charged particle collisions in a low density plasma

Collisions between particles in a low density plasma are described by the Fokker–Planck collision operator. In applications, this nonlinear integro-differential operator is often approximated by linearised or ad-hoc model operators due to computational cost and complexity. In this work, we present an implementation of the nonlinear Fokker–Planck collision operator written in terms of Rosenbluth potentials in the Rosenbluth–MacDonald–Judd (RMJ) form. The Rosenbluth potentials may be obtained either by direct integration or by solving partial differential equations (PDEs) similar to Poisson's equation: we optimise for performance and scalability by using sparse matrices to solve the relevant PDEs. We represent the distribution function using a tensor-product continuous-Galerkin finite-element representation and we derive and describe the implementation of the weak form of the collision operator. We present tests demonstrating a successful implementation using an explicit time integrator and we comment on the speed and accuracy of the operator. Finally, we speculate on the potential for applications in the current and next generation of kinetic plasma models.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Active operator learning with predictive uncertainty quantification for partial differential equations

With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. Here, we propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework’s uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings.

97 MATHEMATICS AND COMPUTING↗