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.

翻译：受近期物理信息神经网络（PINNs）成功求解玻尔兹曼型方程[S. Jin, Z. Ma, and K. Wu, J. Sci. Comput., 94 (2023), pp. 57]的启发，本文对PINNs在全局麦克斯韦分布附近逼近玻尔兹曼方程解的过程进行了严格的误差分析。主要挑战源于速度空间无界域中定义的非局部二次相互作用项。在无界域上分析该相互作用项需要引入截断函数，这要求采用精细的分析技术。作为该分析的推广，我们还证明了基于微宏观分解神经网络所具有的渐近保持特性。