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March 28, 2026

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

Key Points

The research aims to rigorously analyze error estimates of physics-informed neural networks (PINNs) in solving Boltzmann equations.
Conducted error analysis of PINNs applied to Boltzmann equations near a global Maxwellian.
Incorporated a truncation function to handle nonlocal interactions in velocity space.
Provided proof of the asymptotic preserving property for micro-macro decomposition-based neural networks.
Established rigorous error estimates for PINNs approximating the Boltzmann equation.
Demonstrated how the truncation function influences the error analysis in an unbounded domain.
Validated the asymptotic preserving property for the proposed neural network framework.

Abstract

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.

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Cite This Study

Abdo et al. (Mon,) studied this question.

synapsesocial.com/papers/69c772d98bbfbc51511e344b https://doi.org/https://doi.org/10.1093/imanum/draf142

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