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

Abstract Motivated by the recent successful application of physics-informed neural networks (PINNs) to solve Boltzmann-type equations (Jin, S., Ma, Z., & Wu, K. (2023), Asymptotic-preserving neural networks for multiscale time-dependent linear transport equations. J. Sci. Comput., 94, 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.