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Results for “universal differential equations”

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 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Theoretical modeling of hepatitis C acute infection in liver-humanized mice support pre-clinical assessment of candidate viruses for controlled-human-infection studies

Designing and carrying out a controlled human infection (CHI) model for hepatitis C virus (HCV) is critical for vaccine development. However, key considerations for a CHI model protocol include understanding of the earliest viral-host kinetic events during the acute phase and susceptibility of the viral isolate under consideration for use in the CHI model to antiviral treatment before any infections in human volunteers can take place. Humanized mouse models lack adaptive immune responses but provide a unique opportunity to obtain quantitative understanding of early HCV kinetics and develop mathematical models to further understand viral and innate immune response dynamics during acute HCV infection. We show that the models reproduce the measured HCV kinetics in humanized mice, which are consistent with early acute HCV-host dynamics in immunocompetent chimpanzees. Our findings suggest that humanized mice are well-suited to support development of a CHI model. In-silico and in-vivo modeling estimates provide a starting point to characterize candidate viruses for testing in CHI model studies.

Agent-based modeling↗
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 OSTI2025

Application of a temporal multiscale method for efficient simulation of degradation in PEM Water Electrolysis under dynamic operating conditions

Hydrogen is emerging as a vital energy carrier, driven by the need to reduce carbon emissions. Proton Electrolyte Membrane Water Electrolysis (PEMWE) enables hydrogen production under fluctuating renewable power conditions but requires improved understanding and stability of the anode catalyst layer under dynamic operating conditions, especially with low noble metal loadings. Long-term degradation experiments are both time-consuming and costly; therefore, a systematic, model-aided approach is essential. In the present work, a temporal multiscale method is applied to reduce the computational effort of simulating long-term degradation processes in PEMWE, with an exemplary focus on catalyst dissolution. A mechanistic model incorporating the oxygen evolution reaction, catalyst dissolution, and hydrogen permeation from the cathode to the anode was hypothesized and implemented. In this way, the local periodicity of transport and reaction processes in dynamic PEMWE operation, which influence the gradual degradation of the catalyst layer, is captured. The temporal multiscale method significantly reduces the computational effort of simulation, decreasing processing time from hours to mere minutes. This efficiency gain is attributed to the limited evolution of Slow-Scale variables during each period of time P of the Fast-Scale variables. Consequently, simulation is required only until local periodicity is achieved within each Slow-Scale time step. Hence, the fully resolved dynamic problem is decoupled into these two scales, employing a heterogeneous multiscale technique. The developed approach effectively accelerates parameter estimation and predictive simulations, supporting systematic modeling of PEMWE degradation under dynamic conditions.

08 HYDROGEN↗
NASA NTRS1996

Liquid Annular Seal Research

This paper highlights the accomplishments on a joint effort between NASA - Marshall Space Flight Center and Texas A and M University to develop accurate seal analysis software for use in rocket turbopump design, design audits and trouble shooting. Results for arbitrary clearance profile, transient simulation, thermal effects solution and flexible seal wall model are presented. A new solution for eccentric seals based on cubic spline interpolation and ordinary differential equation integration is also presented.

Palazzolo, Alan B.↗
DOE OSTI2025

Hardware acceleration for HPS algorithms in two and three dimensions

We provide a flexible, open-source framework for hardware acceleration, namely massively-parallel execution on general-purpose graphics processing units (GPUs), applied to the hierarchical Poincaré–Steklov (HPS) family of algorithms for building fast direct solvers for linear elliptic partial differential equations. To take full advantage of the power of hardware acceleration, we propose two variants of HPS algorithms to improve performance on two- and three-dimensional problems. In the two-dimensional setting, we introduce a novel recomputation strategy that minimizes costly data transfers to and from the GPU; in three dimensions, we modify and extend the adaptive discretization technique of Geldermans and Gillman [1] to greatly reduce peak memory usage. We provide an open-source implementation of these methods written in JAX, a high-level accelerated linear algebra package, which allows for the first integration of a high-order fast direct solver with automatic differentiation tools. We conclude with extensive numerical examples showing our methods are fast and accurate on two- and three-dimensional problems.

Fast direct solvers↗
DOE OSTI2025

Extended Galerkin Neural Network Approximation of Singular Variational Problems with Error Control

We present extended Galerkin neural networks, a variational framework for approximating general boundary value problems (BVPs) with error control. The main contributions of this work are (1) a rigorous theory guiding the construction of new weighted least squares variational formulations suitable for use in neural network approximation of general BVPs, and (2) an “extended” feedforward network architecture which incorporates and is even capable of learning singular solution structures, thus greatly improving approximability of singular solutions. Furthermore, numerical results are presented for several problems, including steady Stokes flow around reentrant corners and in convex corners with Moffatt eddies in order to demonstrate efficacy of the method.

a posteriori error estimate↗
DOE OSTI2025

Nonintrusive projection-based reduced order modeling using stable learned differential operators

Nonintrusive projection-based reduced order models (ROMs) are essential for dynamics prediction in multi-query applications where underlying governing equations are known but the access to the source of the underlying full order model (FOM) is unavailable; that is, FOM is a glass-box. This article proposes a learn-then-project approach for nonintrusive model reduction. In the first step of this approach, high-dimensional stable sparse learned differential operators (S-LDOs) are determined using the generated data. In the second step, the ordinary differential equations, comprising these S-LDOs, are used with suitable dimensionality reduction and low-dimensional subspace projection methods to provide equations for the evolution of reduced states. This approach allows easy integration into the existing intrusive ROM framework to enable nonintrusive model reduction while allowing the use of Petrov–Galerkin projections. The applicability of the proposed approach is demonstrated for Galerkin and LSPG projection-based ROMs through four numerical experiments: 1-D scalar advection, 1-D Burgers, 2-D scalar advection and 1-D scalar advection–diffusion–reaction equations. In conclusion, the results indicate that the proposed nonintrusive ROM strategy provides accurate and stable dynamics prediction.

42 ENGINEERING↗
DOE OSTI2025

Neural Active Manifolds: Nonlinear Dimensionality Reduction for Uncertainty Quantification

We present a new approach for nonlinear dimensionality reduction, specifically designed for computationally expensive mathematical models. We leverage autoencoders to discover a one-dimensional neural active manifold (NeurAM) capturing the model output variability, through the aid of a simultaneously learnt surrogate model with inputs on this manifold. Our method only relies on model evaluations and does not require the knowledge of gradients. The proposed dimensionality reduction framework can then be applied to assist outer loop many-query tasks in scientific computing, like sensitivity analysis and multifidelity uncertainty propagation. In particular, we prove, both theoretically under idealized conditions, and numerically in challenging test cases, how NeurAM can be used to obtain multifidelity sampling estimators with reduced variance by sampling the models on the discovered low-dimensional and shared manifold among models. Several numerical examples illustrate the main features of the proposed dimensionality reduction strategy and highlight its advantages with respect to existing approaches in the literature.

Autoencoders↗
DOE OSTI2025

Generative AI models for learning flow maps of stochastic dynamical systems in bounded domains

Simulating stochastic differential equations (SDEs) in bounded domains, presents significant computational challenges due to particle exit phenomena, which requires accurate modeling of interior stochastic dynamics and boundary interactions. Despite the success of machine learning-based methods in learning SDEs, existing learning methods are not applicable to SDEs in bounded domains because they cannot accurately capture the particle exit dynamics. We present a unified hybrid data-driven approach that combines a conditional diffusion model with an exit prediction neural network to capture both interior stochastic dynamics and boundary exit phenomena. Our ML model consists of two major components: a neural network that learns exit probabilities using binary cross-entropy loss with rigorous convergence guarantees, and a training-free diffusion model that generates state transitions for non-exiting particles using closed-form score functions. The two components are integrated through a probabilistic sampling algorithm that determines particle exit at each time step and generates appropriate state transitions. Here, the performance of the proposed approach is demonstrated via three test cases: a one-dimensional simplified problem for theoretical verification, a two-dimensional advection-diffusion problem in a bounded domain, and a three-dimensional problem of interest to magnetically confined fusion plasmas.

Bounded domains↗
NASA NTRS1990

Efficient parallel solution of parabolic equations - Implicit methods on the Cedar multicluster

A class of implicit methods for the parallel solution of linear parabolic differential equations based on Pade and Chebyshev rational approximations to the matrix exponential are presented. It is pointed out that this approach incorporates both natural hierarchical parallelism, improved intrinsic efficiency, and fewer timesteps. These advantages lead to an extremely fast family of methods for the solution of certain time-dependent problems. These techniques are illustrated with numerical experiments on the University of Illinois Cedar multicluster architecture. The experiments indicate that implicit methods of very high degree offer great promise for the solution of certain parabolic problems when in computational environment with parallel resources. Hierarchically organized parallel computers, such as the Cedar multicluster, are found to be especially attractive for these schemes.

Gallopoulos, E.↗
DOE OSTI2025

What can solve the strong CP problem?

Three possible strategies have been advocated to solve the strong CP problem. The first is the axion, a dynamical mechanism that relaxes any initial value of the CP violating angle $\overline{θ}$ to zero. The second is the imposition of new symmetries that are believed to set $\overline{θ}$ to zero in the UV. The third is the acceptance of the fine tuning of parameters. We argue that the latter two solutions do not solve the strong CP problem. The θ term of QCD is not a parameter — it does not exist in the Hamiltonian. Rather, it is a property of the quantum state that our universe finds itself in, arising from the fact that there are CP violating states of a CP preserving Hamiltonian. It is not eliminated by imposing parity as a symmetry since the underlying theory is already parity symmetric and that does not preclude the existence of CP violating states. Moreover, since the value of θ realized in our universe is a consequence of measurement, it is inherently random and cannot be fine tuned by choice of parameters. Rather any fine tuning would require a tuning between parameters in the theory and the random outcome of measurement. Our results considerably strengthen the case for the existence of the axion and axion dark matter. The confusion around θ arises from the fact that unlike classical mechanics, the Hamiltonian and Lagrangian are not equivalent in quantum mechanics. The Hamiltonian defines the differential time evolution, whereas the Lagrangian is a solution to this evolution. Consequently, initial conditions could in principle appear in the Lagrangian but not in the Hamiltonian. This results in aspects of the initial condition such as θ misleadingly appearing in the Lagrangian as parameters. We comment on the similarity between the θ vacua and the violations of the constraint equations of classical gauge theories in quantum mechanics.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

From clutter to clarity: Emergent neural operators via questionnaire metrics

Real-world datasets in chemical engineering and bioengineering processes—such as those from catalytic reactors, multiphase flows, polymerization reactors, bioreactors, and clinical trials—can often be unlabeled or disorganized, rendering the training of existing supervised learning models ineffective at learning the underlying dynamics. To salvage these datasets for decision-making, we first seek to obtain clarity from the cluttered data. Here, we present a framework for developing “structural” generative models, discovering emergent equations, and constructing efficient emulators from scrambled datasets by integrating unsupervised organizational learning techniques (Questionnaires) with advanced deep learning architectures (Deep Hidden Physics Models and Deep Operator Networks). Our approach is demonstrated on two illustrative model systems: (a) a 1D advection–diffusion partial differential equation representing a winding underground pipe and (b) an ensemble of Stuart–Landau oscillators, an agent-based system of coupled ordinary differential equations. In both cases, we successfully reconstruct meaningful spatial, temporal, and parameter embeddings from scrambled data, enabling good predictions of system dynamics. As a result, we highlight the framework’s potential for broader applications, enabling data-driven system identification in fields with inherently disorganized or hidden parameter spaces.

42 ENGINEERING↗
DOE OSTI2025

Assimilating partial observation to enhance feedback control of stochastic dynamical systems

Here, in this paper, we present a novel methodology to tackle feedback optimal control problems in scenarios where the exact state of the controlled process is unknown. It integrates data assimilation techniques and optimal control solvers to manage partial observation of the state process, a common occurrence in practical scenarios. Traditional stochastic optimal control methods assume full state observation, which is often not feasible in real-world fluid dynamics control problems. Our approach underscores the significance of utilizing observational data to inform control policy design. Specifically, we introduce a kernel learning backward stochastic differential equation (SDE) filter to enhance data assimilation efficiency and propose a sample-wise stochastic optimization method within the stochastic maximum principle framework. We demonstrate the efficacy and accuracy of our method in the control of advection-diffusion-reaction flow problem and the Dubins airplane maneuvering problem with model uncertainty.

data driven↗
DOE OSTI2024

Extending the Takagi-Taupin equations for x-ray nanobeam Bragg coherent diffraction

We present an approach for simulating x-ray nanobeam Bragg coherent diffraction patterns based on the Takagi-Taupin equations. Compared to conventional methods, the current approach can be universally applied to any weakly strained system including semi-infinite crystals that diffract dynamically. It addresses issues such as the curved wave front and redivergence of the focused incident beam. In conclusion, we show excellent agreement with experimental data for a strained La 0.7 ⁢Sr 0.3⁢ MnO 3 thin film on a SrTiO 3 substrate and a path to extracting physical information using automatic differentiation.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS1994

Dynamo generation of magnetic field in the white dwarf GD 358

On the basis of Whole Earth Telescope observations of the g-mode oscillation spectrum of the white dwarf GD 358, Winget et al. find evidence for significant differential rotation and for a time-varying magnetic field concentrated in the surface layers of this star. Here we argue on theoretical grounds that this magnetic field is produced by an alpha omega dynamo operating in the lower part of a surface convection zone in GD 358. Our argument is based on numerical solutions of the nonlinear, local dynamo equations of Robinson & Durney, with specific parameters based on our detailed models of white-dwarf convective envelopes, and universal constants determined by a calibration with the the Sun's dynamo. The calculations suggest a dynamo cycle period of about 6 years for the fundamental mode, and periods as short as 1 year for the higher-order modes that are expected to dominate in view of the large dynamo number we estimate for GD 358. These dynamo periods are consistent with the changes in the magnetic field of GD 358 over the span of 1 month inferred by Winget et. al. from their observations. Our calculations also suggest a peak dynamo magnetic field strength at the base of the surface convection zone of about 1800 G, which is consistent with the field strength inferred from the observations.

Markiel, J. Andrew↗
DOE OSTI2025

Learning Physically Interpretable Atmospheric Models From Data With WSINDy

The multiscale and turbulent nature of Earth's atmosphere has historically rendered accurate weather modeling a hard problem. Recently, there has been an explosion of interest surrounding data-driven approaches to weather modeling, which in many cases show improved forecasting accuracy and computational efficiency when compared to traditional methods. However, many of the current data-driven approaches employ highly parameterized neural networks, often resulting in uninterpretable models and limited gains in scientific understanding. In this work, we address the interpretability problem by explicitly discovering partial differential equations governing atmospheric phenomena, identifying symbolic mathematical models with direct physical interpretations. The purpose of this paper is to demonstrate that, in particular, the weak-form sparse identification of nonlinear dynamics (WSINDy) algorithm can learn effective atmospheric models from both simulated and assimilated data. Our approach adapts the standard WSINDy algorithm to work with high-dimensional fluid data of arbitrary spatial dimension.

58 GEOSCIENCES↗
NASA NTRS1996

Comparative Study of Advanced Turbulence Models for Turbomachinery

A computational study has been undertaken to study the performance of advanced phenomenological turbulence models coded in a modular form to describe incompressible turbulent flow behavior in two dimensional/axisymmetric and three dimensional complex geometry. The models include a variety of two equation models (single and multi-scale k-epsilon models with different near wall treatments) and second moment algebraic and full Reynolds stress closure models. These models were systematically assessed to evaluate their performance in complex flows with rotation, curvature and separation. The models are coded as self contained modules that can be interfaced with a number of flow solvers. These modules are stand alone satellite programs that come with their own formulation, finite-volume discretization scheme, solver and boundary condition implementation. They will take as input (from any generic Navier-Stokes solver) the velocity field, grid (structured H-type grid) and computational domain specification (boundary conditions), and will deliver, depending on the model used, turbulent viscosity, or the components of the Reynolds stress tensor. There are separate 2D/axisymmetric and/or 3D decks for each module considered. The modules are tested using Rocketdyn's proprietary code REACT. The code utilizes an efficient solution procedure to solve Navier-Stokes equations in a non-orthogonal body-fitted coordinate system. The differential equations are discretized over a finite-volume grid using a non-staggered variable arrangement and an efficient solution procedure based on the SIMPLE algorithm for the velocity-pressure coupling is used. The modules developed have been interfaced and tested using finite-volume, pressure-correction CFD solvers which are widely used in the CFD community. Other solvers can also be used to test these modules since they are independently structured with their own discretization scheme and solver methodology. Many of these modules have been independently tested by Professor C.P. Chen and his group at the University of Alabama at Huntsville (UAH) by interfacing them with own flow solver (MAST).

Hadid, Ali H.↗
DOE OSTI2024

Difference equations and integral families for Witten diagrams

We show that tree-level and one-loop Mellin space correlators in anti-de Sitter space obey certain difference equations, which are the direct analog to the differential equations for Feynman loop integrals in the flat space. Finite-difference relations, which we refer to as “summation-by-parts relations”, in parallel with the integration-by-parts relations for Feynman loop integrals, are derived to reduce the integrals to a basis. We illustrate the general methodology by explicitly deriving the difference equations and summation-by-parts relations for various tree-level and one-loop Witten diagrams up to the four-point bubble level.

AdS-CFT Correspondence↗