Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures

TL;DR

This paper develops a machine learning-based moment closure model for the radiative transfer equation that enforces global hyperbolicity, ensuring stability and accuracy over long times, with promising results on benchmark tests.

Contribution

It introduces a method to enforce hyperbolicity in ML closure models by seeking a symmetrizer, improving stability and physical consistency.

Findings

Model inherits dissipativeness of RTE

Preserves correct diffusion limit as Knudsen number approaches zero

Demonstrates good accuracy and stability on benchmark tests

Abstract

This is the second paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation (RTE). In our previous work \cite{huang2021gradient}, we proposed an approach to directly learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional $P_N$ closure. However, the ML moment closure model in \cite{huang2021gradient} is not able to guarantee hyperbolicity and long time stability. We propose in this paper a method to enforce the global hyperbolicity of the ML closure model. The main idea is to seek a symmetrizer (a symmetric positive definite matrix) for the closure system, and derive constraints such that the system is globally symmetrizable hyperbolic. It is shown that the new ML closure system inherits the dissipativeness of the RTE and preserves the correct diffusion limit as the Knunsden number goes to zero. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, long time stability and generalizability of our globally hyperbolic ML closure model.