Search NASA⌕ Search

SEARCH · Search NASA

Results for “equation”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Solving high-dimensional partial integral differential equations: The finite expression method

Partial integro-differential equations (PIDEs) have broad applications in the sciences, from electro-magnetism to options pricing. Here, in this paper, we introduce a new finite expression method (FEX) to solve PIDEs. This approach builds upon the original FEX and its inherent advantages with new advances: 1) A novel method of parameter grouping is proposed to reduce the number of coefficients in high-dimensional function approximation; 2) A Taylor series approximation method is implemented to significantly improve the computational efficiency and accuracy of the evaluation of the integral terms of PIDEs. The new FEX based method, denoted FEX-PG to indicate the addition of the parameter grouping (PG) step to the algorithm, provides both high accuracy and interpretable numerical solutions, with the outcome being an explicit equation that facilitates intuitive understanding of the underlying solution structures. These features are often absent in traditional methods, such as finite element methods (FEM) and finite difference methods, as well as in deep learning-based approaches. To benchmark our method against recent advances, we apply the new FEX-PG to solve benchmark PIDEs in the literature. In high-dimensional settings, FEX-PG exhibits strong and robust performance, achieving relative errors on the order of single precision machine epsilon, significantly outperforming existing approaches based on neural networks.

Combinatorial optimization↗
DOE OSTI2026

A score-based diffusion model approach for adaptive learning of stochastic partial differential equation solutions

In this paper, we propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central role in modeling complex physical systems under uncertainty, but their numerical solutions often suffer from model errors and reduced accuracy due to incomplete physical knowledge and environmental variability. To address these challenges, we encode the governing physics into the score function of a diffusion model using simulation data and incorporate observational information via a likelihood-based correction in a reverse-time stochastic differential equation. This enables adaptive learning through iterative refinement of the solution as new data becomes available. To improve computational efficiency in high-dimensional settings, we introduce the ensemble score filter, a training-free approximation of the score function designed for real-time inference. Numerical experiments on benchmark SPDEs demonstrate the accuracy and robustness of the proposed method under sparse and noisy observations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Quantum mechanical closure of partial differential equations with symmetries

We develop a statistical framework for the dynamical closure of spatiotemporal dynamics governed by partial differential equations. Employing the mathematical framework of quantum mechanics to embed the original classical dynamics into a quantum mechanical representation, we use the space of quantum density operators to model the unresolved degrees of freedom of the original dynamics in a statistical sense, and the framework of quantum measurement to predict their contributions to the resolved dynamics. The embedded dynamics is discretized by a positivity preserving process, leading to a compressed representation that is invariant under the dynamical symmetries of the resolved dynamics. We present a data based formulation of the closure scheme and apply it to a closure problem for the shallow water equations. The numerical results demonstrate that our closure model can accurately predict the main features of the true dynamics, including for out of sample initial conditions.

Delay embedding↗
DOE OSTI2025

Analytic soliton solutions of nonlinear extensions of the Schrödinger equation

A method is presented to construct analytic solitary wave solutions in nonlinear extensions of the Schrödinger equation starting from analytic solutions of the ordinary Schrödinger equation. We provide several examples illustrating the method. We rederive three well-known soliton solutions including the N-dimensional non-relativistic Gausson as well as the one-dimensional 1 / cosh-soliton and a theory with a power-like nonlinearity proportional to |ψ| 2λ with λ > 0. We also find several new solutions in different nonlinear theories in various space dimensions which, to the best of our knowledge, have not yet been discussed in literature. Our method can be used to construct further nonlinear theories and generalized to relativistic soliton theories, and may have many applications.

Analytical soliton solutions↗
DOE OSTI2025

Stability analysis of the Eulerian–Lagrangian finite volume methods for nonlinear hyperbolic equations in one space dimension

In this paper, we construct a novel Eulerian–Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine–Hugoniot junmp condition. Unfortunately, to avoid the intersection of the partition lines, the time step sizes are still limited. To fix this gap, we detect effective troubled cells (ETCs) and carefully design the influence region of each ETC, within which the partitioned space-time regions are merged together to form a new one. Then with the new partition of the space-time domain, we theoretically prove that the proposed first-order scheme with Euler forward time discretization is total-variation-diminishing and maximum-principle-preserving with at least twice larger time step constraints than the classical first order Eulerian method for Burgers’ equation. Numerical experiments verify the optimality of the designed time step sizes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Sensitivity of an integrated experiment to uncertainty in the high explosive equations of state

Traditionally, hydrodynamics simulations are performed with a single equation-of-state (EOS) to describe each material. These EOSs typically have a physics-informed functional form with adjustable parameters that are calibrated in order to replicate small-scale data. However, because the calibration data have uncertainty and there are typically inherent degeneracies in fitting the EOS, there are actually multiple EOSs that might be consistent with calibration data. In this work, we perform uncertainty quantification (UQ) for the reactant and product equations of state for the high explosive PBX 9501 to yield an ensemble of EOSs that match the uncertain small-scale calibration data. We then simulate an experiment of an explosively formed penetrator repeatedly with different EOSs to both validate the UQ analysis and determine the effects of EOS uncertainty on the prediction of quantities of interest in the experiment. In general, we find good agreement between the simulation predictions and the experimental measurements, and we identify an EOS variable that contributes most directly to the spread in the predictions as the EOSs are varied.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Equation-of-motion internally contracted multireference unitary coupled-cluster theory

The accurate computation of excited states remains a challenge in electronic structure theory, especially for systems with a ground state that requires a multireference treatment. In this work, we introduce a novel equation-of-motion (EOM) extension of the internally contracted multireference unitary coupled-cluster framework (ic-MRUCC), termed EOM-ic-MRUCC. EOM-ic-MRUCC follows the transform-then-diagonalize approach, in analogy to its non-unitary counterpart. By employing a projective approach to optimize the ground state, the method retains additive separability and proper scaling with system size. We show that excitation energies are size-intensive if the EOM operator satisfies the “killer” and the projective conditions. Furthermore, we propose to represent changes in the reference state upon electron excitation via projected many-body operators that span the active orbitals and show that the EOM equations formulated in this way are invariant with respect to active orbital rotations. We test the EOM-ic-MRUCC method truncated to single and double excitations by computing the potential energy curves for several excited states of a BeH2 model system, the HF molecule, and water undergoing symmetric dissociation. Across these systems, our method delivers accurate excitation energies and potential energy curves within 5 mE h (∼0.14 eV) from full configuration interaction. Here, we find that truncating the Baker–Campbell–Hausdorff series to fourfold commutators contributes negligible errors (on the order of 10 −5 E h or less), offering a practical route to highly accurate excited-state calculations with reduced computational overhead.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2025

Direct nonparametric multimessenger constraints on the equation of state of cold dense nuclear matter

We utilize the now substantial amount of astrophysical observations of neutron stars (NSs), along with perturbative quantum chromodynamics (pQCD) calculations at high density, to directly constrain the NS equation of state (EOS). To this end, we construct nonparametric EOS priors by using Gaussian processes trained on 75 EOSs, which include models with either hadrons, hyperons, or quarks at high densities. We create a prior using the full EOS sample (model agnostic), and one prior for each EOS family to test model discrimination. We introduce a novel inference approach, which allows the simultaneous sampling of intrinsic and extrinsic parameters of binary NS mergers, as well as a nonparametric equation of state. We showcase this method in a Bayesian updating scheme by first performing a complete analysis of the binary NS merger event GW170817 with minimal assumptions, and sequentially adding information from x-ray and radio NS observations, along with pQCD calculations. Besides providing standard constraints, such as the pressure at twice nuclear saturation density 𝑝⁡(2⁢𝜌 sat ) = 4.3$^{+0.6}_{−0.6}$ × 10 34 dyne/cm 2 , at 95% confidence level, for the model agnostic prior, our methodology shows how the choice of EOS families used in conditioning changes the inferred astrophysical properties of the EOS, namely tidal deformability and maximum supported NS mass. We find hyperonic priors predicting higher tidal deformabilities for a 1.4⁢𝑀 ⊙ NS, and hadronic priors being preferred by the considered astrophysical data.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Four-dimensional QCD equation of state with multiple chemical potentials

Here, we construct a four-dimensional version of the equation of state (EoS) model neos, neos-4d, as a function of the temperature and chemical potentials of baryon, electric charge, and strangeness for the hot and dense quantum chromodynamics (QCD) matter created in relativistic nuclear collisions. This EoS enables multiple conserved charge current evolution in a relativistic fluid. Input from lattice QCD simulations and a hadron resonance gas model is considered for constructing the equation of state. We investigate its applicability to the relativistic hydrodynamic description of nuclear collisions and present a method for efficient numerical implementation.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Finite density QCD equation of state: Critical point and lattice-based 𝑇′ expansion

Here, we present a novel construction of the QCD equation of state (EoS) at finite baryon density. Our work combines a recently proposed resummation scheme for lattice QCD results with the universal critical behavior at the QCD critical point. This allows us to obtain a family of equations of state in the range 0 ≤ 𝜇 𝐵 ≤ 700 MeV and 25 MeV ≤ 𝑇 ≤ 800 MeV, which match lattice QCD results near 𝜇 𝐵 =0 while featuring a critical point in the 3D Ising model universality class. The position of the critical point can be chosen within the range accessible to beam-energy scan heavy-ion collision experiments. The strength of the singularity and the shape of the critical region are parametrized using a standard parameter set. We impose stability and causality constraints and discuss the available ranges of critical point parameter choices, finding that they extend beyond earlier parametric QCD EoS proposals. We present thermodynamic observables, including baryon density, pressure, entropy density, energy density, baryon susceptibility and speed of sound, that cover a wide range in the QCD phase diagram relevant for experimental exploration.

Astronomy & Astrophysics↗
DOE OSTI2026

Exact solution of the DeWitt-Brehme-Hobbs equation in copropagating electromagetic and gravitational waves

An accelerated charge interacts with its own electromagnetic field, a phenomenon known as electromagnetic radiation reaction. The DeWitt-Brehme-Hobbs (DWBH) equation describes the motion of a charged mass in the presence of combined electromagnetic and gravitational fields, taking into account electromagnetic radiation-reaction effects. Here, we find the first exact analytical solution of the DWBH equation in the case of a charged mass in the presence of copropagating and otherwise arbitrary electromagnetic and gravitational plane waves. As a consequence of the Penrose limit, the scenario considered here can be seen as a local limit around ultrarelativistic trajectories in a general curved spacetime. Finally, the paradigmatic example of an electromagnetic wave in the presence of a constant-amplitude gravitational wave is worked out explicitly and it is shown how the presence of the gravitational wave can qualitatively change electromagnetic radiation-reaction effects.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Fully quantum algorithm for mesoscale fluid simulations with application to partial differential equations

Fluid flow simulations marshal our most powerful computational resources. In many cases, even this is not enough. Quantum computers provide an opportunity to speed up traditional algorithms for flow simulations. We show that lattice-based mesoscale numerical methods can be executed as efficient quantum algorithms due to their statistical features. This approach revises a quantum algorithm for lattice gas automata to reduce classical computations and state preparation at every time step. For this, the algorithm approximates the qubit relative phases and subtracts them at the end of each time step. Phases are evaluated using the iterative phase estimation algorithm and subtracted using single-qubit rotation phase gates. Further, this method optimizes the quantum resource required and makes it more appropriate for near-term quantum hardware. We also demonstrate how the checkerboard deficiency that the D1Q2 scheme presents can be resolved using the D1Q3 scheme. The algorithm is validated by simulating two canonical partial differential equations: the diffusion and Burgers' equations on different quantum simulators. We find good agreement between quantum simulations and classical solutions for the presented algorithm.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Asymptotic Relaxation of Moment Equations for a Multi-species, Homogeneous BGK Model

Multi-species BGK models describe the dynamics of rarefied gases with constituent particles of different elements or compounds with potentially nontrivial velocity distributions. Here, in this paper, moment equations for the bulk velocities, energies, and temperatures of a spatially homogeneous multi-species BGK model are examined. A key challenge in analyzing these equations is the fact that the collision frequencies are allowed to depend on the species temperatures, which allows for more realistic simulations of dilute gas flow. Therefore, a positive lower bound is established for the species temperatures. With this lower bound, a global existence and uniqueness of solutions to the coupled velocity-energy ODE system is established. The lower bound also enables a proof of exponential decay to a unique steady-state solution. Numerical results are presented to demonstrate how the bulk velocities and temperatures relax for large times.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Power System Frequency Dynamics Modeling, State Estimation, and Control using Neural Ordinary Differential Equations (NODEs) and Soft Actor-Critic (SAC) Machine Learning Approaches

With the global energy transition of the electric power system, grid control, supervision, and protection is becoming more challenging. With the increasing integration of renewable energy sources (RES), the system dynamics are changing, causing traditional power system dynamic modeling with swing equation-based modeling approaches to fail. Additionally, the converter-dominated power grid is decreasing the system inertia, making the power system more fragile to the frequency swings. This paper first investigates and compares the application of a model-based Kalman filter state estimation approach with (i) a model-free machine learning approach --- neural ordinary differential equations (NODEs) --- and (ii) a data-driven system identification (SysId) approach to model and infer critical state values of the power system frequency dynamics. Then a model predictive control (MPC) framework is compared to a model-free Soft Actor-Critic (SAC) reinforcement learning (RL) control algorithm in providing efficient fast frequency response (FFR) to the power system frequency dynamics. The approaches are compared in terms of their performance goals as well as their per-timestep computational efficiency. Furthermore, the comparative study for state estimation shows that for the model-free requirement, both NODEs and SysId can provide accurate state estimates; however, with increasing model complexity, NODEs can be a better choice for model identification. Similarly, the results from the FFR comparative study show that the SAC RL-based FFR, once trained, outperforms MPC with better control signals and faster computation time, making the SAC RL-based FFR better option for providing FFR to the power system.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Reduced basis approximations of parameterized dynamical partial differential equations via neural networks

Projection-based reduced order models are effective at approximating parameter-dependent differential equations that are parametrically separable. When parametric separability is not satisfied, which occurs in both linear and nonlinear problems, projection-based methods fail to adequately reduce the computational complexity. Devising alternative reduced order models is crucial for obtaining efficient and accurate approximations to expensive high-fidelity models. In this work, we develop a timestepping procedure for dynamical parameter-dependent problems, in which a neural-network is trained to propagate the coefficients of a reduced basis expansion. This results in an online stage with a computational cost independent of the size of the underlying problem. Here, we demonstrate our method on several parabolic partial differential equations, including a problem that is not parametrically separable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Comparison of integral equations used to study ${T}_{cc}^{+}$ for a stable D *

We perform a detailed comparison between three formalisms used in recent studies of DD* scattering at heavier-than-physical pion masses, which aim to understand the properties of the doubly-charmed tetraquark, ${T}_{cc}^{+}$ (3875). These methods are the three-particle relativistic field theory (RFT) formalism, the two-body Lippmann-Schwinger (LS) equation with chiral effective field theory potentials, and the two-particle relativistic framework proposed by Baião Raposo and Hansen (BRH approach). In a simplified single-channel setting, we derive the conditions under which the infinite-volume integral equations from the RFT and BRH approaches reduce to the LS form. We present numerical examples showing that differences between these methods can be largely removed by adjusting short-range couplings. We also address a number of technical issues in the RFT approach.

Hadronic Spectroscopy↗
DOE OSTI2025

Data-driven equation-free dynamics applied to many-protein complexes: The microtubule tip relaxation

Microtubules (MTs) constitute the largest components of the eukaryotic cytoskeleton and play crucial roles in various cellular processes, including mitosis and intracellular transport. The property allowing MTs to cater to such diverse roles is attributed to dynamic instability, which is coupled to the hydrolysis of GTP (guanosine-5'-triphosphate) to GDP (guanosine-5'-diphosphate) within the β-tubulin monomers. Understanding the equilibrium dynamics and the structural features of both GDP- and GTP-complexed MT tips, especially at an all-atom level, remains challenging for both experimental and computational methods because of their dynamic nature and the prohibitive computational demands of simulating large, many-protein systems. This study employs the “equation-free” multiscale computational method to accelerate the relaxation of all-atom simulations of MT tips toward their putative equilibrium conformation. Using large MT lattice systems (14 protofilaments × 8 heterodimers) comprising ~21-38 million atoms, we applied this multiscale approach to leapfrog through time and nearly double the computational efficiency in realizing relaxed all-atom conformations of GDP- and GTP-complexed MT tips. Commencing from an initial 4 μs unbiased all-atom simulation, we interleave coarse projective “equation-free” jumps with short bursts of all-atom molecular dynamics simulation to realize an additional effective simulation time of 1.875 μs. Our 5.875 μs of effective simulation trajectories for each system expose the subtle yet essential differences in the structures of MT tips as a function of whether β-tubulin monomer is complexed with GDP or GTP, as well as the lateral interactions within the MT tip, offering a refined understanding of features underlying MT dynamic instability. Furthermore, the approach presents a robust and generalizable framework for future explorations of large biomolecular systems at atomic resolution.

Wu, Jiangbo [University of Chicago, IL (United Sta↗
DOE OSTI2025

Jacobian-based model diagnostics and application to equation oriented modeling of a carbon capture system

It can be difficult to identify the specific variables or equations responsible for convergence issues in large mathematical programming models. The Institute for the Design of Advanced Energy Systems Integrated Platform (IDAES-IP) contains a tool to identify poorly scaled constraints and variables by searching for rows and columns of the Jacobian matrix with small L2-norms. A singular value decomposition is then performed to identify degenerate sets of equations and remaining scaling issues. Here, this work presents a flowsheet developed for post-combustion carbon capture using a monoethanolamine (MEA) solvent system as a case study. This work takes the reader through the entire process of model diagnostics and reformulation, from a basic introduction to the mathematics behind these model diagnostics to the reformulations necessary to make the model numerically robust, including a significantly modified enhancement factor model.

IDAES↗