DATA-DRIVEN, STRUCTURE-PRESERVING APPROXIMATIONS TO ENTROPY-BASED MOMENT CLOSURES FOR KINETIC EQUATIONS

William A. Porteous; Ming Tse P. Laiu; Cory D. Hauck

doi:10.4310/CMS.2023.v21.n4.a1

DATA-DRIVEN, STRUCTURE-PRESERVING APPROXIMATIONS TO ENTROPY-BASED MOMENT CLOSURES FOR KINETIC EQUATIONS

William A. Porteous, Ming Tse P. Laiu, Cory D. Hauck

Multiscale Methods

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

Abstract

We present a data-driven approach for approximating entropy-based closures of moment systems from kinetic equations. The proposed closure learns the entropy function by fitting the map between the moments and the entropy of the moment system, and thus does not depend on the spacetime discretization of the moment system or specific problem configurations such as initial and boundary conditions. With convex and C2 approximations, this data-driven closure inherits several structural properties from entropy-based closures, such as entropy dissipation, hyperbolicity, and H-Theorem. We construct convex approximations to the Maxwell-Boltzmann entropy using convex splines and neural networks, test them on the plane source benchmark problem for linear transport in slab geometry, and compare the results to the standard, entropy-based systems which solve a convex optimization problem to find the closure. Numerical results indicate that these data-driven closures provide accurate solutions in much less computation time than that required by the optimization routine.

Original language	English
Pages (from-to)	885-913
Number of pages	29
Journal	Communications in Mathematical Sciences
Volume	21
Issue number	4
DOIs	https://doi.org/10.4310/CMS.2023.v21.n4.a1
State	Published - 2023

Funding

∗Received: August 17, 2021; Accepted (in revised form): August 19, 2022. Communicated by Pierre Degond. This work is sponsored by 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 the 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).

Funders	Funder number
U.S. Department of Energy	DE-AC05-00OR22725
Advanced Scientific Computing Research

Keywords

Entropy based moment closures
datadriven approximations
kinetic equations
machine learning
radiative transfer equations
structure preserving methods

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title = "DATA-DRIVEN, STRUCTURE-PRESERVING APPROXIMATIONS TO ENTROPY-BASED MOMENT CLOSURES FOR KINETIC EQUATIONS",

abstract = "We present a data-driven approach for approximating entropy-based closures of moment systems from kinetic equations. The proposed closure learns the entropy function by fitting the map between the moments and the entropy of the moment system, and thus does not depend on the spacetime discretization of the moment system or specific problem configurations such as initial and boundary conditions. With convex and C2 approximations, this data-driven closure inherits several structural properties from entropy-based closures, such as entropy dissipation, hyperbolicity, and H-Theorem. We construct convex approximations to the Maxwell-Boltzmann entropy using convex splines and neural networks, test them on the plane source benchmark problem for linear transport in slab geometry, and compare the results to the standard, entropy-based systems which solve a convex optimization problem to find the closure. Numerical results indicate that these data-driven closures provide accurate solutions in much less computation time than that required by the optimization routine.",

keywords = "Entropy based moment closures, datadriven approximations, kinetic equations, machine learning, radiative transfer equations, structure preserving methods",

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AU - Laiu, Ming Tse P.

AU - Hauck, Cory D.

PY - 2023

Y1 - 2023

N2 - We present a data-driven approach for approximating entropy-based closures of moment systems from kinetic equations. The proposed closure learns the entropy function by fitting the map between the moments and the entropy of the moment system, and thus does not depend on the spacetime discretization of the moment system or specific problem configurations such as initial and boundary conditions. With convex and C2 approximations, this data-driven closure inherits several structural properties from entropy-based closures, such as entropy dissipation, hyperbolicity, and H-Theorem. We construct convex approximations to the Maxwell-Boltzmann entropy using convex splines and neural networks, test them on the plane source benchmark problem for linear transport in slab geometry, and compare the results to the standard, entropy-based systems which solve a convex optimization problem to find the closure. Numerical results indicate that these data-driven closures provide accurate solutions in much less computation time than that required by the optimization routine.

AB - We present a data-driven approach for approximating entropy-based closures of moment systems from kinetic equations. The proposed closure learns the entropy function by fitting the map between the moments and the entropy of the moment system, and thus does not depend on the spacetime discretization of the moment system or specific problem configurations such as initial and boundary conditions. With convex and C2 approximations, this data-driven closure inherits several structural properties from entropy-based closures, such as entropy dissipation, hyperbolicity, and H-Theorem. We construct convex approximations to the Maxwell-Boltzmann entropy using convex splines and neural networks, test them on the plane source benchmark problem for linear transport in slab geometry, and compare the results to the standard, entropy-based systems which solve a convex optimization problem to find the closure. Numerical results indicate that these data-driven closures provide accurate solutions in much less computation time than that required by the optimization routine.

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DATA-DRIVEN, STRUCTURE-PRESERVING APPROXIMATIONS TO ENTROPY-BASED MOMENT CLOSURES FOR KINETIC EQUATIONS

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