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# Communications in Mathematical Sciences

## Volume 21 (2023)

### Number 4

### Data-driven, structure-preserving approximations to entropy-based moment closures for kinetic equations

Pages: 885 – 913

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n4.a1

#### Authors

#### 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 $C^2$ 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, kinetic equations, radiative transfer equations, datadriven approximations, structure-preserving methods, machine learning

#### 2010 Mathematics Subject Classification

35Q20, 65M99, 82C40, 82C70

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.

Received 17 August 2021

Received revised 5 May 2022

Accepted 19 August 2022

Published 24 March 2023