Machine learning-based moment closure model for the linear Boltzmann equation with uncertainties

TL;DR

This paper introduces a machine learning-based moment closure model for the linear Boltzmann equation, effectively handling both deterministic and stochastic cases with improved stability and accuracy.

Contribution

It develops a neural network approach to learn the unclosed moments, ensuring hyperbolicity and stability, and extends it to stochastic problems using gPC-based stochastic Galerkin methods.

Findings

Demonstrates the effectiveness of the ML-based closure through numerical experiments.

Ensures system hyperbolicity and stability via derived constraints.

Achieves accurate results for both deterministic and stochastic linear Boltzmann equations.

Abstract

The Boltzmann equation, a fundamental equation in kinetic theory, serves as a bridge between microscopic particle dynamics and macroscopic continuum mechanics. However, deriving closed macroscopic moment systems from the Boltzmann equation remains a long-standing challenge due to the intrinsic non-closure of the moment hierarchy. In this paper, we propose a machine learning (ML)-based moment closure model for the linear Boltzmann equation, addressing both the deterministic and stochastic settings. Our approach leverages neural networks to learn the spatial gradient of the unclosed highest-order moment, enabling effective training through natural output normalization. For the deterministic problem, to ensure global hyperbolicity and stability, we derive and apply the constraints that enforce symmetrizable hyperbolicity of the system. For the stochastic problem, we adopt the generalized polynomial chaos (gPC)-based stochastic Galerkin method to discretize the random variables, resulting in a system for which the approach in the deterministic case can be used similarly. Several numerical experiments are shown to demonstrate the effectiveness and accuracy of our ML-based moment closure model for the linear Boltzmann equation with or without uncertainties.