Error estimates of physics-informed neural networks for approximating Boltzmann equation
Numerical Analysis
2025-11-20 v3 Numerical Analysis
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.
Cite
@article{arxiv.2407.08383,
title = {Error estimates of physics-informed neural networks for approximating Boltzmann equation},
author = {Elie Abdo and Lihui Chai and Ruimeng Hu and Xu Yang},
journal= {arXiv preprint arXiv:2407.08383},
year = {2025}
}