Error estimates of physics-informed neural networks for approximating Boltzmann equation

TL;DR

This paper provides a rigorous error analysis of physics-informed neural networks (PINNs) for solving the Boltzmann equation, addressing challenges from nonlocal interactions and unbounded velocity domains, and demonstrating asymptotic preserving properties.

Contribution

It offers the first detailed error estimates for PINNs applied to the Boltzmann equation, including handling unbounded domains and nonlocal terms, and proves asymptotic preserving properties.

Findings

Error estimates for PINNs in Boltzmann equation approximation

Handling of nonlocal quadratic interaction terms on unbounded domains

Proof of asymptotic preserving property with micro-macro decomposition

Abstract

Motivated by the recent successful application of physics-informed neural networks (PINNs) to solve Boltzmann-type equations [S. Jin, Z. Ma, and K. Wu, J. Sci. Comput., 94 (2023), pp. 57], we provide a rigorous error analysis for PINNs in approximating the solution of the Boltzmann equation near a global Maxwellian. The challenge arises from the nonlocal quadratic interaction term defined in the unbounded domain of velocity space. Analyzing this term on an unbounded domain requires the inclusion of a truncation function, which demands delicate analysis techniques. As a generalization of this analysis, we also provide proof of the asymptotic preserving property when using micro-macro decomposition-based neural networks.