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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

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↗
NASA NTRS1977

Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE's

If a linear ordinary differential equation with polynomial coefficients is converted into integrated form then the formal substitution of a Chebyshev series leads to recurrence equations defining the Chebyshev coefficients of the solution function. An explicit formula is presented for the polynomial coefficients of the integrated form in terms of the polynomial coefficients of the differential form. The symmetries arising from multiplication and integration of Chebyshev polynomials are exploited in deriving a general recurrence equation from which can be derived all of the linear equations defining the Chebyshev coefficients. Procedures for deriving the general recurrence equation are specified in a precise algorithmic notation suitable for translation into any of the languages for symbolic computation. The method is algebraic and it can therefore be applied to differential equations containing indeterminates.

Geddes, K. O.↗
NASA NTRS1979

An implicit factored scheme for the compressible Navier-Stokes equations. II - The numerical ODE connection

An attempt is made to establish a connection between linear multistep methods for applications to ordinary differential equations and their extension (by approximate factorization) to alternating direction implicit methods for partial differential equations. An earlier implicit factored scheme for the compressible Navier-Stokes equations is generalized by innovations that (1) increase the class of temporal difference schemes to include all linear multistep methods, (2) optimize the class of unconditionally stable factored schemes by a new choice of unknown variable, and (3) improve the computational efficiency by the introduction of quasi-one-leg methods.

Beam, R. M.↗
NASA NTRS1982

Wafering insight provided by the ODE method

The orientation-dependent etching method of slicing was investigated to see if it could be used to form more slices from high quality silicon crystals than can be achieved by mechanical slicing methods. Orientation-dependent slicing uses preferential etching down narrow slots in a silicon slab on form slices. The method has several possible advantages including high slicing yield (sq m/kg), plane parallel, thin slices, ready for processing and the chance of high throughput and low costs. There are limitations in the need for simple crystals, and in restricted depth of slicing.

Soclof, S. I.↗
NASA NTRS1981

Modified Runge-Kutta methods for solving ODES

A class of Runge-Kutta formulas is examined which permit the calculation of an accurate solution anywhere in the interval of integration. This is used in a code which seldom has to reject a step; rather it takes a reduced step if the estimated error is too large. The absolute stability implications of this are examined.

Vanvu, T.↗
NASA NTRS1984

Ode to gravity: Depth/diameter for fresh craters on Mercury

Opinions conflict over the role of surface gravity in shaping impact craters on Mercury. One view holds that the effects of g are evident in measurable aspects of crater form; other investigators find little or no evidence for g's geomorphic importance. Ambiguity in the role of g and other variables in cratering on Mercury stems largely from uncertainty in identifying major geomorphic contrasts and the crater sizes at which they occur. One of these, depth/diameter (d/D), undergoes a major change at the transition from simple (bowl shaped) to complex (peaks and terraces) crater interiors. Four least-squares d/D fits for fresh craters on Mercury were attemped. The results are inconsistent. The d/D data that should resolve previous shortcomings is presented. The revised d/D distributions for simple and complex craters, which intersect at a diameter of about 5 km, support the initial thesis that g substantially influences the form of Mercury's craters.

Pike, R. J.↗
NASA NTRS1990

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.

Yee, H. C.↗
NASA NTRS1990

Linear iterative solvers for implicit ODE methods

The numerical solution of stiff initial value problems, which lead to the problem of solving large systems of mildly nonlinear equations are considered. For many problems derived from engineering and science, a solution is possible only with methods derived from iterative linear equation solvers. A common approach to solving the nonlinear equations is to employ an approximate solution obtained from an explicit method. The error is examined to determine how it is distributed among the stiff and non-stiff components, which bears on the choice of an iterative method. The conclusion is that error is (roughly) uniformly distributed, a fact that suggests the Chebyshev method (and the accompanying Manteuffel adaptive parameter algorithm). This method is described, also commenting on Richardson's method and its advantages for large problems. Richardson's method and the Chebyshev method with the Mantueffel algorithm are applied to the solution of the nonlinear equations by Newton's method.

Saylor, Paul E.↗
NASA NTRS1994

Nonlinear Asymptotic Integration Algorithms for One-dimensional Autonomous Dissipative First-order Odes

Nonlinear asymptotic integrators are applied to one-dimensional, nonlinear, autonomous, dissipative, ordinary differential equations. These integrators, including a one-step explicit, a one-step implicit, and a one- and two-step midpoint algorithm, are designed to follow the asymptotic behavior of a system approaching a steady state. The methods require that the differential equation be written in a particular asymptotic form. This is always possible for a one-dimensional equation with a globally asymptotic steady state. In this case, conditions are obtained to guarantee that the implicit algorithms are well defined. Further conditions are determined for the implicit methods to be contractive. These methods are all first order accurate, while under certain conditions the midpoint algorithms may also become second order accurate. The stability of each method is investigated and an estimate of the local error is provided.

Haslach, Henry W., Jr.↗
NASA NTRSDate not supplied

Discontinuous Galerkin and Related Methods for ODE

Starting from the standard integral formulation, the DG method is derived here in differential form. The key ingredient is a polynomial called the correction function, which helps ‘correct’ the discontinuous solution by approximating the jump and yields a continuous one. Under the right Radau quadrature, this continuous solution is identical to the solutions by the right Radau collocation and the continuous Galerkin (CG) methods. Next, the correction function facilitates the construction of the associated implicit Runge-Kutta schemes (IRK-DG). Different quadratures for DG result in different IRK-DG methods: left Radau quadrature in Radau IA, right Radau quadrature in Radau IIA or right Radau collocation, and Gauss quadrature in a method called DG-Gauss. The construction of IRK-DG clarifies the meaning and facilitates the proofs of various 𝐵(𝑝), 𝐶(𝜂), and 𝐷(𝜁) conditions for accuracy. The two consequences of these conditions are that all 𝑠-stage IRK-DG methods are accurate to order 2𝑠− 1, and the IRK-DG methods of Radau type are unique. Numerical examples showing the behavior of the DG solutions are provided. In all, the correction function plays a key role and helps establish the relations among the DG, IRK DG, collocation, and CG methods.

Numerical Methods for Ordinary Differential Equati↗
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, ↗