Error estimates of physics-informed neural networks for approximating Boltzmann equation
<|reference_start|>Error estimates of physics-informed neural networks for approximating Boltzmann equation: 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.<|reference_end|>
@article{abdo2024error,
    title={Error estimates of physics-informed neural networks for approximating
  Boltzmann equation},
    author={Elie Abdo, Lihui Chai, Ruimeng Hu, Xu Yang},
    journal={arXiv preprint arXiv:2407.08383},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.08383},
    primaryClass={math.NA cs.NA}
}