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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 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

A Data-Driven Method for Modeling Creep-Fatigue Stress- Strain Behavior Using Neural ODEs

In this paper, we introduce a data-driven machine learning approach for modeling one-dimensional stress–strain behavior under cyclic loading, utilizing experimental data from the nickel-based Alloy 617. The study employs uniaxial creep–fatigue test data acquired under various loading histories and compares two distinct neural network-based ODE models. The first model, known as the black-box model, comprehensively describes the strain–stress relationship using a Neural ODE equation. To interpret this black-box model, we apply the Sparse Identification of Nonlinear Dynamical Systems (SINDy) technique, transforming the black-box model into an equation-based model using symbolic regression. The second model, the Neural flow rule model, incorporates Hooke’s Law for the linear elastic component, with the nonlinear part characterized by a Neural ODE. Both models are trained with experimental data to accurately reflect the observed stress–strain behavior. We conduct a detailed comparison with the standard Chaboche model, which includes three back stresses. Our results demonstrate that the neural network-based ODE models precisely capture the experimental creep–fatigue mechanical behavior, exceeding the standard Chaboche model’s accuracy. Furthermore, an interpretable model derived from the black-box neural ODE model through symbolic regression achieves accuracy comparable to the Chaboche model, enhancing its interpretability. The results highlight the potential of neural network-based ODE models to depict complex creep–fatigue behavior, eliminating the necessity for experts to define a specific, material-focused model form.

creep-fatigue↗
DOE OSTI2024

Implementation and (Inverse Modified) Error Analysis for Implicitly Templated ODE-Nets

We focus on learning unknown dynamics from data using ODE-nets templated on implicit numerical initial value problem solvers. First, we perform inverse modified error analysis of the ODE-nets using unrolled implicit schemes for ease of interpretation. It is shown that training an ODE-net using an unrolled implicit scheme returns a close approximation of an inverse modified differential equation (IMDE). In addition, we establish a theoretical basis for hyperparameter selection when training such ODE-nets, whereas current strategies usually treat numerical integration of ODE-nets as a black box. We thus formulate an adaptive algorithm which monitors the level of error and adapts the number of (unrolled) implicit solution iterations during the training process, so that the error of the unrolled approximation is less than the current learning loss. This helps accelerate training while maintaining accuracy. Several numerical experiments are performed to demonstrate the advantages of the proposed algorithm compared to nonadaptive unrollings and validate the theoretical analysis. Here, we also note that this approach naturally allows for incorporating partially known physical terms in the equations, giving rise to what is termed “gray box” identification.

ODE-nets↗
DOE OSTI2025

Computing Nonlinear Power Spectra Across Dynamical Dark Energy Model Space with Neural ODEs

I show how to compute the nonlinear power spectrum across the entire $w(z)$ dynamical dark energy model space. Using synthetic ΛCDM data, I train a neural ordinary differential equation (ODE) to infer the evolution of the nonlinear matter power spectrum as a function of the background expansion and mean matter density across ∼9 Gyr of cosmic evolution. After training, the model generalises to any dynamical dark energy model parameterised by $w(z)$. With little optimisation, the neural ODE is accurate to within 4% up to $k = 5\, h\, {\mathrm Mpc}^{−1}$. Unlike simulation rescaling methods, neural ODEs naturally extend to summary statistics beyond the power spectrum that are sensitive to the growth history.

cosmology↗
DOE OSTI2026

A stiff order condition theory for Runge–Kutta methods applied to semilinear ODEs

Classical convergence theory of Runge–Kutta methods assumes that the time step is small relative to the Lipschitz constant of the ordinary differential equation (ODE). For stiff problems, that assumption is often violated, and a problematic degradation in accuracy, known as order reduction, can arise. Methods with high stage order, e.g., Gauss–Legendre and Radau, are known to avoid order reduction, but they must be fully implicit. For the broad class of semilinear ODEs, which consist of a stiff linear term and non-stiff nonlinear term, we show that weaker conditions suffice. Here, our new semilinear order conditions are formulated in terms of orthogonality relations and can be enumerated by rooted trees. Finally, we prove global error bounds that hold uniformly with respect to stiffness of the linear term.

Mathematics and Computing↗
DOE OSTI2025

Acceleration of Power System Dynamic Simulations Using a Deep Equilibrium Layer and Neural ODE Surrogate

The dominant paradigm for power system dynamic simulation is to build system-level simulations by combining physics-based models of individual components. The sheer size of the system along with the rapid integration of inverter-based resources exacerbates the computational burden of running time domain simulations. Here, in this paper, we propose a data-driven surrogate model based on implicit machine learningspecifically deep equilibrium layers and neural ordinary differential equationsto learn a reduced order model of a portion of the full underlying system. The data-driven surrogate achieves similar accuracy and reduction in simulation time compared to a physics-based surrogate, without the constraint of requiring detailed knowledge of the underlying dynamic models. This work also establishes key requirements needed to integrate the surrogate into existing simulation workflows; the proposed surrogate is initialized to a steady state operating point that matches the power flow solution by design.

Neural ordinary differential equations↗
DOE OSTI2025

Understanding latent timescales in neural ordinary differential equation models of advection-dominated dynamical systems

The neural ordinary differential equation (ODE) framework has shown considerable promise in recent years in developing highly accelerated surrogate models for complex physical systems characterized by partial differential equations (PDEs). For PDE-based systems, state-of-the-art neural ODE strategies leverage a two-step procedure to achieve this acceleration: a nonlinear dimensionality reduction step provided by an autoencoder, and a time integration step provided by a neural-network based model for the resultant latent space dynamics (the neural ODE). This work explores the applicability of such autoencoder-based neural ODE strategies for PDEs in which advection terms play a critical role. More specifically, alongside predictive demonstrations, physical insight into the sources of model acceleration (i.e., how the neural ODE achieves its acceleration) is the scope of the current study. Such investigations are performed by quantifying the effects of both autoencoder and neural ODE components on latent system time-scales using eigenvalue analysis of dynamical system Jacobians. To this end, the sensitivity of various critical training parameters – de-coupled versus end-to-end training, latent space dimensionality, and the role of training trajectory length, for example – to both model accuracy and the discovered latent system timescales is quantified. Furthermore, this work specifically uncovers the key role played by the training trajectory length (the number of rollout steps in the loss function during training) on the latent system timescales: larger trajectory lengths correlate with an increase in limiting neural ODE time-scales, and optimal neural ODEs are found to recover the largest time-scales of the full-order (ground-truth) system. Demonstrations are performed across fundamentally different unsteady fluid dynamics configurations influenced by advection: (1) the Kuramoto–Sivashinsky equations (2) Hydrogen-Air channel detonations (the compressible reacting Navier–Stokes equations with detailed chemistry), and (3) 2D Atmospheric flow.

Advection-dominated dynamical systems↗
DOE OSTI2024

Implementing Ordinary Differential Equation Solvers in Rust Programming Language for Modeling Vehicle Powertrain Systems: Preprint

Efficient and accurate ordinary differential equation (ODE) solvers are necessary for powertrain and vehicle dynamics modeling. However, current commercial ODE solvers can be financially prohibitive, leading to a need for accessible, effective, open-source ODE solvers designed for powertrain modeling. Rust is a compiled programming language that has the potential to be used for fast and easy-to-use powertrain models, given its exceptional computational performance, robust package ecosystem, and short time required for modelers to become proficient. However, of the three commonly used (>3,000 downloads) packages in Rust with ODE solver capabilities, only one has more than four numerical methods implemented, and none are designed specifically for modeling physical systems. Therefore, the goal of the Differential Equation System Solver (DESS) was to implement accurate ODE solvers in Rust designed for the component-based problems often seen in powertrain modeling. DESS is a text-based software package that provides a flexible framework for building and solving systems of ODEs. This allows DESS to be included as a dependency for automotive powertrain models that require a variety of solvers and solver configurations. Seven explicit ODE solver methods have been implemented in DESS: Euler’s, Heun’s, midpoint, Ralston’s, classic Runge-Kutta, Bogacki-Shampine, and Cash-Karp. These represent five fixed-step methods and two adaptive-step methods. This paper shows that the solver implementations increase accuracy and computational efficiency compared to Euler's method when modeling a system of three thermal masses in Rust. DESS also includes features designed for modeling component-based physical systems. Users can define relationships between nodes in their system, which the package then translates into a system of equations, leading to simpler and more intuitive code. In the case of a three-thermal-mass system, the user can specify node thermal properties (e.g., thermal capacitance), how nodes are interconnected, and thermal conductance between nodes rather than providing a system of equations. The core contribution from this work is an open-source, text-based Rust package with ODE solvers for automotive powertrain modeling to support cost-free, fast, and accurate simulation.

ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2025

Active Learning of Microgrid Frequency Dynamics Using Neural Ordinary Differential Equations

Accurate frequency modelling of inverter‐based resource (IBR)‐dominated power systems is crucial for ensuring stable, reliable and resilient operations, particularly given their inherent low‐inertia characteristics and fast dynamics that traditional swing equation‐based models inadequately capture. This paper explores neural ordinary differential equations (Neural ODEs) as a computationally efficient, data‐driven framework for modelling power system frequency dynamics, specifically within microgrids integrating high penetrations of distributed energy resources (DERs). The developed neural ODEs framework incorporates a neural network architecture designed to capture input dynamics. By actively perturbing the system with a known signal, the Python‐based neural ODEs framework was trained using measured system states and inputs, without the need for detailed system information. The framework, tested on a model of the Cordova, AK, microgrid, achieved a goodness of fit ranging from 60% to 99% across different state variables and maintained a mean square error in the 10 -6 p.u. range under square and step excitation signals. The proposed approach demonstrated robustness to measurement noise and initial condition variations while maintaining low computational complexity suitable for real‐time power system control applications. Furthermore, transfer learning enabled the neural ODEs model to adapt to the following changes in system topology or generator dispatch, highlighting its effectiveness for dynamic microgrids with frequently evolving configurations and diverse DERs.

Aryal, Tara [South Dakota State Univ., Brookings, ↗
DOE OSTI2024

Multiphysics Time-Integration for Turbulent Combustion at the Exascale

Turbulent reacting flow systems are often modeled with coupled time-dependent partial differential equations (PDEs). Solving such equations can easily tax the world's largest supercomputers. One pragmatic strategy for attacking such problems is to split the PDEs into components that can more easily be solved in isolation. This generic operator-splitting strategy leads to a set of ordinary differential equations (ODEs) that need to be solved as part of an "outer-loop" time-stepping approach. In many combustion applications, the ODEs to be solved can be very stiff, exhibiting timescales that span many orders of magnitude. The SUNDIALS library provides a plethora of robust time integration algorithms for solving these ODEs on exascale-capable computing hardware, yet for many complex applications (such multicomponent fuels or emissions predictions), the chemical models remain too complex to solve using reasonable resources. The Quasi-Steady State Approximation (QSSA) can be an effective tool for reducing the size and stiffness of the simulations. In this talk, I will discuss the use of the SUDIALS library of ODE solvers together with automatic code generation tools to solve complex turbulent reacting flow problems using QSSA models.

chemistry↗
DOE OSTI2024

Machine learning for the identification of phase transitions in interacting agent-based systems: A Desai-Zwanzig example

Deriving closed-form analytical expressions for reduced-order models, and judiciously choosing the closures leading to them, has long been the strategy of choice for studying phase- and noise-induced transitions for agent-based models (ABMs). In this paper, we propose a data-driven framework that pinpoints phase transitions for an ABM—the Desai-Zwanzig model—in its mean-field limit, using a smaller number of variables than traditional closed-form models. To this end, we use the manifold learning algorithm Diffusion Maps to identify a parsimonious set of data-driven latent variables, and we show that they are in one-to-one correspondence with the expected theoretical order parameter of the ABM. We then utilize a deep learning framework to obtain a conformal reparametrization of the data-driven coordinates that facilitates, in our example, the identification of a single parameter-dependent ordinary differential equation (ODE) in these coordinates. Additionally, we identify this ODE through a residual neural network inspired by a numerical integration scheme (forward Euler). We then use the identified ODE—enabled through an odd symmetry transformation—to construct the bifurcation diagram exhibiting the phase transition.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

The Method of Finite Averages

The Method of Finite Averages (MoFA) is a rigorous multiscale modeling methodology for efficiently modeling multi-physical phenomena in heterogeneous porous media. The code developed in this project aims to perform the numerical calculations required to formulate, implement, and verify MoFA models for Earth and Energy systems (i.e., model verification refers to performing fully-resolved simulations of the systems and comparing their results to those of the models). In general, MoFA transforms partial differential equations (PDEs) describing the fine-scale physics of a system into coupled ordinary differential equations (ODEs)---in time---that describe the coarse-scale---or "average"---physical behaviors of the system. This transformation significantly expedites system simulation, as the coarse-scale ODEs involve vastly fewer degrees of freedom than the fine-scale PDEs. The code developed under this project will allow users to 1.) generate system geometries and numerical meshes, 2.) solve the PDE and ODE systems required for MoFA model formulation and implementation, 3.) solve the PDE systems required to obtain fully-resolved simulation results for model verification, and 4.) compare and plot results (e.g., the model and fully-resolved simulation solutions, the error between the solutions, etc.).

Pietrzyk, KyleM [Lawrence Livermore National Labor↗
DOE OSTI2024

SUNDIALS time integrators for exascale applications with many independent systems of ordinary differential equations

Many complex systems can be accurately modeled as a set of coupled time-dependent partial differential equations (PDEs). However, solving such equations can be prohibitively expensive, easily taxing the world’s largest supercomputers. One pragmatic strategy for attacking such problems is to split the PDEs into components that can more easily be solved in isolation. This operator splitting approach is used ubiquitously across scientific domains, and in many cases leads to a set of ordinary differential equations (ODEs) that need to be solved as part of a larger “outer-loop” time-stepping approach. The SUNDIALS library provides a plethora of robust time integration algorithms for solving ODEs, and the U.S. Department of Energy Exascale Computing Project (ECP) has supported its extension to applications on exascale-capable computing hardware. In this paper, we highlight some SUNDIALS capabilities and its deployment in combustion and cosmology application codes (Pele and Nyx, respectively) where operator splitting gives rise to numerous, small ODE systems that must be solved concurrently.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Improving ADAM through an implicit-explicit (IMEX) time-stepping approach

The ADAM optimizer, often used in machine learning for neural network training, corresponds to an underlying ordinary differential equation (ODE) in the limit of very small learning rates. Here, this work shows that the classical ADAM algorithm is a first-order implicit-explicit (IMEX) Euler discretization of the underlying ODE. Employing the time discretization point of view, we propose new extensions of the ADAM scheme obtained by using higher-order IMEX methods to solve the ODE. Based on this approach, we derive a new optimization algorithm for neural network training that performs better than classical ADAM on several regression and classification problems.

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↗