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

Steffen Schotthöfer; Paul Laiu; Martin Frank; Cory D. Hauck

doi:10.1016/j.jcp.2025.113967

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

Steffen Schotthöfer, Paul Laiu, Martin Frank, Cory D. Hauck

Multiscale Methods

Research output: Contribution to journal › Article › peer-review

Abstract

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 GitHub¹^,².

Original language	English
Article number	113967
Journal	Journal of Computational Physics
Volume	533
DOIs	https://doi.org/10.1016/j.jcp.2025.113967
State	Published - Jul 15 2025

Funding

This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).The work of Steffen Schotth\u00F6fer and Martin Frank has been funded by the Priority Programme \u201CTheoretical Foundations of Deep Learning (SPP2298)\u201D by the Deutsche Forschungsgemeinschaft. The work of Steffen Schotth\u00F6fer, Cory Hauck and Paul Laiu is sponsored by the Applied Mathematics Program at the Office of Advanced Scientific Computing Research, U.S. Department of Energy, and performed at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). The work of Steffen Schotth\u00F6fer and Martin Frank has been funded by the Priority Programme \u201CTheoretical Foundations of Deep Learning (SPP2298)\u201D by the Deutsche Forschungsgemeinschaft. The work of Steffen Schotth\u00F6fer, Cory Hauck and Paul Laiu is sponsored by the Applied Mathematics Program at the Office of Advanced Scientific Computing Research, U.S. Department of Energy, and performed at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

Keywords

Entropy closure
Kinetic theory
Moment methods
Neural networks
Regularized optimization

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10.1016/j.jcp.2025.113967

Cite this

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

abstract = "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 GitHub1,2.",

keywords = "Entropy closure, Kinetic theory, Moment methods, Neural networks, Regularized optimization",

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T1 - Structure-preserving neural networks for the regularized entropy-based closure of a linear, kinetic, radiative transport equation

AU - Schotthöfer, Steffen

AU - Laiu, Paul

AU - Frank, Martin

AU - Hauck, Cory D.

PY - 2025/7/15

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N2 - 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 GitHub1,2.

AB - 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 GitHub1,2.

KW - Entropy closure

KW - Kinetic theory

KW - Moment methods

KW - Neural networks

KW - Regularized optimization

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DO - 10.1016/j.jcp.2025.113967

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JF - Journal of Computational Physics

M1 - 113967

ER -

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

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