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Results for “Linear Systems of 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 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

PIAFS: A 2D nonlinear hydrodynamics code to model gaseous optics

The survivability of final optics is expected to be a major challenge for all future inertial fusion energy concepts. Due to their higher damage threshold, gaseous optics have been identified as a promising solution to this problem. Gaseous optics can be created through the photoabsorption of spatially modulated UV light, which induces various chemical processes that heat the gas. This heating leads to a pressure perturbation, which in turn launches a density perturbation that can imprint a refractive index modulation such as a grating. In this article, we introduce a parallel C/C++ code to simulate gaseous optics. PIAFS2D is a high-order conservative finite-difference code to solve the compressible Navier–Stokes equations along with the photochemical heating sources on Cartesian grids. The simulations are validated by the linear theory derived in a previous paper [Michel et al., Phys. Rev. Appl. 22, 024014 (2024)]. For larger perturbations, the behavior of the system—particularly the evolution of the generated acoustic wave—demonstrates strong nonlinearity. PIAFS2D allows the study of nonlinear behaviors and can be used for the design of high-efficiency gaseous optics elements in realistic experimental conditions.

Oudin, A. [Lawrence Livermore National Laboratory ↗
DOE OSTI2025

Structure-preserving neural networks for the regularized entropy-based closure of a linear, kinetic, radiative transport equation

The main challenge of large-scale numerical simulation of radiation transport is the high memory and computation time requirements of discretization methods for kinetic equations. In this work, we derive and investigate a neural network-based approximation to the entropy-based closure method to accurately compute the solution of the multi-dimensional moment system with a low memory footprint and competitive computational time. We extend methods developed for the standard entropy-based closure to the regularized entropy-based closures. The main idea is to interpret structure-preserving neural network approximations of the regularized entropy-based closure as a two-stage approximation to the original entropy-based closure. We conduct a numerical analysis of this approximation and investigate optimal parameter choices. Our numerical experiments demonstrate that the method has a much lower memory footprint than traditional methods with competitive computation times and simulation accuracy. The code and all trained networks are provided on GitHub.

entropy closure↗
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

Flow dynamics and heat transfer in simplified battery energy storage systems with heated battery modules

Large-scale energy storage systems (ESSs) composed of batteries show promise in addressing current energy challenges, but dissipation of generated heat is important. Here, this paper focuses on buoyant convective flows in simplified ESS battery racks. Natural convection is not generally the primary cooling strategy but can be important in abnormal scenarios where there is module overheat or potentially thermal runaway. We use computational fluid dynamics to investigate the flow dynamics and heat transfer mechanisms in a simplified parameterized rack design. Despite its simplicity, this configuration produces many of the relevant features expected in real ESSs without details of module geometry or hardware, allowing broad conclusions independent of manufacture-specific designs. We start by providing visualizations of the flowfield and measurements of entrainment, heat flux, and pressure. To characterize the dependence on the system parameters, we develop an integral-scale analysis of the average temperature equation to highlight the dominant source terms. We use results from this analysis to derive a steady network model composed of simple algebraic expressions to provide first-order predictions of entrainment through the rack. The network model leads to a linear scaling of the Reynolds number based on convective mass flux with respect to the Grashof number based on the heat source. We deduce empirical relationships that relate the heat exchanged between modules using a surface-averaged Nusselt number as a function of the local Reynolds and Rayleigh numbers. Lastly, we investigate how space between the modules and rack in the spanwise direction creates flow bypass, resulting in different flow pathways.

Battery thermal management↗
DOE OSTI2026

Learning thermodynamic master equations for open quantum systems

The characterization of Hamiltonians and other components of open quantum dynamical systems plays a crucial role in quantum computing and other applications. Scientific machine learning techniques have been applied to this problem in a variety of ways, including by modeling with deep neural networks. However, the majority of mathematical models describing open quantum systems are linear, and the natural nonlinearities in learnable models have not been incorporated using physical principles. We present a data-driven model for open quantum systems that includes learnable, thermodynamically consistent terms. The trained model is interpretable, as it directly estimates the system Hamiltonian and linear components of coupling to the environment. We validate the model on synthetic two and three-level data, as well as experimental two-level data collected from a quantum device at Lawrence Livermore National Laboratory.

Mathematics and Computing↗
DOE OSTI2024

Stochastic Maxwell-Bloch equations for modeling amplified spontaneous emission

An approach for studying atom-radiation interaction has been developed, associating quantum operators with stochastic variables governed by discrete Heisenberg equations. This framework models general multilevel atomic systems using the paraxial approximation in both single- and multipass configurations. Simulation results for x-ray laser oscillators with parameters of interest are presented. Finally, the stochastic model is efficient to solve, as the required computational resources scale linearly with the number of emitters, and it may be applicable to various other quantum systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Predicting the Slowing of Stellar Differential Rotation by Instability-driven Turbulence

Abstract Differentially rotating stars and planets transport angular momentum (AM) internally due to turbulence at rates that have long been a challenge to predict reliably. We develop a self-consistent saturation theory, using a statistical closure approximation, for hydrodynamic turbulence driven by the axisymmetric Goldreich–Schubert–Fricke instability at the stellar equator with radial differential rotation. This instability arises when fast thermal diffusion eliminates the stabilizing effects of buoyancy forces in a system where a stabilizing entropy gradient dominates over the destabilizing AM gradient. Our turbulence closure invokes a dominant three-wave coupling between pairs of linearly unstable eigenmodes and a near-zero frequency, viscously damped eigenmode that features latitudinal jets. We derive turbulent transport rates of momentum and heat and provide them in analytic forms. Such formulae, free of tunable model parameters, are tested against direct numerical simulations; the comparison shows good agreement. They improve upon prior quasi-linear or “parasitic saturation” models containing a free parameter. Given model correspondences, we also extend this theory to heat and compositional transport for axisymmetric thermohaline-instability-driven turbulence in certain regimes.

Astronomy & Astrophysics↗
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 OSTI2025

Current-based metrology with two-terminal mesoscopic conductors

The traditional approach to quantum parameter estimation focuses on the quantum state, deriving fundamental bounds on precision through the quantum Fisher information. In most experimental settings, however, performing arbitrary quantum measurements is highly unfeasible. In open quantum systems, an alternative approach to metrology involves the measurement of stochastic currents flowing from the system to its environment. However, the present understanding of current-based metrology is mostly limited to Markovian master equations. Considering a parameter estimation problem in a two-terminal mesoscopic conductor, we identify the key elements that determine estimation precision within the Landauer-Büttiker formalism. Crucially, this approach allows us to address arbitrary coupling and temperature regimes. Furthermore, we obtain analytical results for the precision in linear-response and zero-temperature regimes. For the specific parameter estimation task that we consider, we demonstrate that the boxcar transmission function is optimal for current-based metrology in all parameter regimes.

Landauer formula↗
DOE OSTI2026

One-Body Properties and Their Perturbative Accuracy with Aufbau Suppressed Coupled Cluster Theory

In this work we derived and implemented the calculation of the one-body reduced density matrix for Aufbau suppressed coupled cluster theory, from which excited state natural orbitals and one-body properties, like atomic populations and dipole moments, are obtained. We utilized the natural orbitals to refine the ASCC solution for simple valence and Rydberg systems, exploring the process of repeatedly solving the ASCC equations in successive natural orbital bases to achieve independence from the starting molecular orbitals. For dipole moments in small molecules where high-level comparison data is available, we find that the accuracy of ASCC essentially matches that of linear response and equation-of-motion coupled cluster as long as care is taken to preserve the response's perturbative completeness.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

An implicit-explicit time splitting strategy for the far SOL plasma fluid model with DG-FEM discretization

We consider a far scrape-off layer (SOL) plasma fluid model of ions that is governed by a Braginskiitype model: a one-dimensional, nonlinear system of advection-diffusion equations coupled with a diffusion equation for neutral particles. Our motivation for studying this system arises from the coupling between the edge plasma and radio-frequency (RF) heating, where solving a far SOL plasma fluid model provides critical insights into edge plasma dynamics. Numerical simulations of plasma fluid models require advanced computational techniques to achieve both efficiency and accuracy, especially when resolving the boundary layer in magnetically confined plasmas. In this work, we propose an implicit-explicit time operator splitting strategy that allows for an efficient solution algorithm, where the diffusive terms are treated semi-implicitly requiring only a linear solve, while the advection part is handled explicitly using a strong-stability-preserving Runge-Kutta (SSP-RK3) scheme. This leads to a fully decoupled system in which the diffusion and advection sub-problems can be solved separately, simplifying the overall solution procedure and allowing for efficient parallelization, which is particularly relevant for exploring the impact of RF heating on the SOL plasma. The main challenge of the discretization is due to the strong coupling between diffusion and advection, particularly through the boundary conditions. This makes implementation of such a scheme in an accurate and stable manner nontrivial. We discuss in detail how to split the equations and manage boundary conditions to maintain stability and well-posedness for each subsystem. We also describe a spatial discretization approach, based on the discontinuous Galerkin finite element method (DG-FEM) and present numerical results for a one-dimensional system.

Burkovska, Olena [ORNL] (ORCID:0000000163101130)↗
DOE OSTI2025

Central Compact Finite‐Difference Scheme With High Spectral Resolution for KdV Equation

This work presents a combination of cell‐node and cell‐centered compact finite difference scheme for the approximation of third derivatives involved in Korteweg–de Vries (KdV) equations. This approach employs a half‐shifted derivative construction at cell centers, avoiding the need for compact interpolation, thereby removing transfer errors; hence, it improves spectral resolution and maintains high‐order accuracy. Fourier analysis is performed to show the spectral properties of the proposed formulation, which provides higher spectral resolutions as compared to node‐based compact schemes. A filtering strategy is incorporated to suppress high‐frequency oscillations without compromising the accuracy of the numerical scheme, and the total variation diminishing Runge Kutta (TVDRK3) method is applied for time integration. Numerical experiments on linear, nonlinear, and coupled KdV systems are conducted, and a comparative analysis with cell‐node compact schemes confirms that the proposed scheme consistently reduces errors by up to an order of magnitude and achieves high spectral resolution properties.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Emergent viscous hydrodynamics from a single quantum particle

We investigate an explicit example of how spatial decoherence can lead to hydrodynamic behavior in the late-time, long-wavelength regime of open quantum systems. We focus on the case of a single nonrelativistic quantum particle linearly coupled to a thermal bath of noninteracting harmonic oscillators at temperature T , a la Caldeira and Leggett. Taking advantage of decoherence in the position representation, we expand the reduced density matrix in powers of the off-diagonal spatial components, so that high-order terms are suppressed at late times. Truncating the resulting power series at second order leads to a set of dissipative transient hydrodynamic equations similar to the nonrelativistic limit of equations widely used in simulations of the quark-gluon plasma formed in ultrarelativistic heavy-ion collisions. Transport coefficients are directly determined by the damping constant γ , which quantifies the influence of the environment. The asymptotic limit of our hydrodynamic equations reduces to the celebrated Navier-Stokes equations for a compressible fluid in the presence of a drag force. Furthermore, our results shed new light on the onset of hydrodynamic behavior in open quantum systems where a system with few degrees of freedom is coupled to a large thermal environment.

Hydrodynamics↗
DOE OSTI2024

Distribution of centrality measures on undirected random networks via the cavity method

The Katz centrality of a node in a complex network is a measure of the node’s importance as far as the flow of information across the network is concerned. For ensembles of locally tree-like undirected random graphs, this observable is a random variable. Its full probability distribution is of interest but difficult to handle analytically because of its “global” character and its definition in terms of a matrix inverse. Leveraging a fast Gaussian Belief Propagation-Cavity algorithm to solve linear systems on tree-like structures, we show that i) the Katz centrality of a single instance can be computed recursively in a very fast way, and ii) the probability P ( K ) that a random node in the ensemble of undirected random graphs has centrality K satisfies a set of recursive distributional equations, which can be analytically characterized and efficiently solved using a population dynamics algorithm. We test our solution on ensembles of Erdős-Rényi and Scale Free networks in the locally tree-like regime, with excellent agreement. The analytical distribution of centrality for the configuration model conditioned on the degree of each node can be employed as a benchmark to identify nodes of empirical networks with over- and underexpressed centrality relative to a null baseline. We also provide an approximate formula based on a rank- 1 projection that works well if the network is not too sparse, and we argue that an extension of our method could be efficiently extended to tackle analytical distributions of other centrality measures such as PageRank for directed networks in a transparent and user-friendly way.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Conformal duality of the nonlinear Schrödinger equation: Theory and applications to parameter estimation

The nonlinear Schrödinger equation (NLSE) in one spatial dimension has stationary solutions similar to those of the linear Schrödinger equation (LSE) as well as more exotic solutions such as solitary waves and quantum droplets. Here, we present a newly discovered conformal duality which unifies the stationary and time-dependent traveling-wave solutions of the one-dimensional cubic-quintic NLSE, the cubic NLSE and LSE. Any two systems that are classified by the same single number called the cross ratio are related by this symmetry. Notably, the conformal duality can also be adapted in Newtonian mechanics and serves as a powerful tool for investigating physical systems that otherwise cannot be directly accessed in experiments. Further, we show that the conformal symmetry is a valuable resource to substantially improve NLSE parameter estimation from noisy empirical data by introducing an optimization afterburner. The new method therefore has far reaching practical applications for nonlinear physical systems. Published by the American Physical Society 2025

Reinhardt, David B. (ORCID:0009000409812838)↗
DOE OSTI2026

Large-scale white noise and cosmology

The generation of white noise on large scales is a generic property of the dynamics of physical systems described by local nonlinear partial differential equations. Nonlinearities prevent the small-scale dynamics from being erased by smoothing. Unresolved small-scale dynamics act as an uncorrelated (white or Poissonian) noise (seemingly stochastic but actually deterministic) contribution to large-scale dynamics. This white noise exists even when the dynamics is very nearly linear. In cases where the power spectrum is sub-Poissonian on large scales, this noise will dominate on the largest scale power no matter the amplitude of the inhomogeneities. Such is the case in the standard model of cosmology, where the primordial density power spectrum is expected to have an almost Harrison-Zel’dovich, P [ k ] ∼ k , spectrum on a much broader range of scales than can be observed. Even though linear gravitational evolution dominates nonlinear corrections by a factor of ∼ 10 5 , the nonobservation of white noise on the Hubble scale precludes the extrapolation of this power law below the comoving 1 pc scale. More generally, observation or nonobservation of large-scale white noise provides a powerful probe of the Universe on very small scales in the early Universe. Gravitational radiation, phase transitions, vorticity, and running of the spectral index are all phenomena that can be probed with large-scale white noise. Large-scale white noise is a nonoptional feature of all cosmological models but one which has not heretofore been appreciated.

Barenboim, Gabriela [Valencia U., IFIC; Valencia U↗
DOE OSTI2024

Bayesian reduced-order deep learning surrogate model for dynamic systems described by partial differential equations

We propose a reduced-order deep-learning surrogate model for dynamic systems described by time-dependent partial differential equations. This method employs space–time Karhunen–Loève expansions (KLEs) of the state variables and space-dependent KLEs of space-varying parameters to identify the reduced (latent) dimensions. Subsequently, a deep neural network (DNN) is used to map the parameter latent space to the state variable latent space. An approximate Bayesian method is developed for uncertainty quantification (UQ) in the proposed KL-DNN surrogate model. The KL-DNN method is tested for the linear advection–diffusion and nonlinear diffusion equations, and the Bayesian approach for UQ is compared with the deep ensembling (DE) approach, commonly used for quantifying uncertainty in DNN models. It was found that the approximate Bayesian method provides a more informative distribution of the PDE solutions in terms of the coverage of the reference PDE solutions (the percentage of nodes where the reference solution is within the confidence interval predicted by the UQ methods) and log predictive probability. The DE method is found to underestimate uncertainty and introduce bias. For the nonlinear diffusion equation, we compare the KL-DNN method with the Fourier Neural Operator (FNO) method and find that KL-DNN is 10% more accurate and needs less training time than the FNO method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Identifying stochastic dynamics via finite expression methods

Modeling stochastic differential equations (SDEs) is crucial for understanding complex dynamical systems in various scientific fields. Recent methods often employ neural network-based models, which typically represent SDEs through a combination of deterministic and stochastic terms. However, these models usually lack interpretability and have difficulty in generalizing beyond their training domain. Here, this paper introduces the Finite Expression Method (FEX), a symbolic learning approach designed to derive interpretable mathematical representations of the deterministic component of SDEs. For the stochastic component, we integrate FEX with advanced generative modeling techniques to provide a comprehensive representation of SDEs. The numerical experiments on linear, nonlinear, and multidimensional SDEs demonstrate that FEX generalizes well beyond the training domain and delivers more accurate long-term predictions compared to neural network-based methods. The symbolic expressions identified by FEX not only improve prediction accuracy but also offer valuable scientific insights into the underlying dynamics of the systems.

Complex dynamical systems↗