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

Elie Abdo, Lihui Chai, Ruimeng Hu, Xu Yang

TL;DR

This work develops a rigorous error-analysis framework for physics-informed neural networks (PINNs) solving the Boltzmann equation near a global Maxwellian, addressing the challenge of the nonlocal collision term on an unbounded velocity space via velocity truncation and a micro-macro PINN approach. It provides generalization and total-error bounds that relate the network loss to the true solution, proves universal approximation-type results for tanh networks, and demonstrates an asymptotic-preserving property for multiscale kinetic models. The analysis hinges on precise local estimates for the nonlocal operators $K$ and $\Gamma$, bilinear controls, and carefully constructed residuals that respect truncated domains. The results establish that, under suitable regularity, vanishing PINN residuals imply convergence to the Boltzmann solution and, in the multiscale limit, convergence to the diffusion equation, with explicit energy estimates ensuring stability. Numerical experiments in a linearized, one-dimensional setting corroborate the theoretical findings, showing that network capacity and depth lead to decreasing errors relative to a spectral reference, and supporting the practical viability of APPINNs for kinetic problems.

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.

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

TL;DR

This work develops a rigorous error-analysis framework for physics-informed neural networks (PINNs) solving the Boltzmann equation near a global Maxwellian, addressing the challenge of the nonlocal collision term on an unbounded velocity space via velocity truncation and a micro-macro PINN approach. It provides generalization and total-error bounds that relate the network loss to the true solution, proves universal approximation-type results for tanh networks, and demonstrates an asymptotic-preserving property for multiscale kinetic models. The analysis hinges on precise local estimates for the nonlocal operators and , bilinear controls, and carefully constructed residuals that respect truncated domains. The results establish that, under suitable regularity, vanishing PINN residuals imply convergence to the Boltzmann solution and, in the multiscale limit, convergence to the diffusion equation, with explicit energy estimates ensuring stability. Numerical experiments in a linearized, one-dimensional setting corroborate the theoretical findings, showing that network capacity and depth lead to decreasing errors relative to a spectral reference, and supporting the practical viability of APPINNs for kinetic problems.

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.
Paper Structure (15 sections, 10 theorems, 118 equations, 2 figures)

This paper contains 15 sections, 10 theorems, 118 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. PINNs for Boltzmann equation
  4. Main results
  5. Related literature
  6. Organization of the paper
  7. Local Estimates for the Nonlocal Operators and
  8. Generalization Error Estimates
  9. Total Error Estimates
  10. Asymptotic preserving property
  11. Residuals and Errors
  12. Preliminary Lemmas.
  13. Asymptotic Preserving Property.
  14. Numerical experiments
  15. Conclusion

Key Result

Proposition 1.1

The collision operator satisfies the following properties:

Figures (2)

  • Figure 1: Interpretation of APNNs reproduced from jin2023asymptotic with slightly modified notations to be consistent with this paper. Here, $\mathcal{F}^\varepsilon$ represents the multiscale equation \ref{['eq:boltzmann_veps']}--\ref{['eq:linearker']}, and $\mathcal{F}^0$ represents the limit macroscopic model (diffusion equation) \ref{['eq:macro-diffusion']}. Let $\mathcal{E}_G(\mathcal{F}^\varepsilon)$ be the generalization error of PINN associated with \ref{['eq:macro-part-bolt']}--\ref{['eq:micro-part-bolt']}. The asymptotic preserving property aims to demonstrate that when $\varepsilon$ is small, the PINN solution obtained by minimizing $\mathcal{E}_G(\mathcal{F}^\varepsilon)$ can accurately approximate $\rho^0$, given that $\mathcal{E}_G(\mathcal{F}^\varepsilon)$ is small.
  • Figure 2: Comparison of the PINN solution and the reference solution for the Boltzmann equation. The left column displays the errors in the distribution functions, while the right column shows the errors in the density functions. From top to bottom, the results correspond to fully-connected neural networks with 2, 3, and 4 layers, each tested with layer widths of 16, 32, 64, 128, 256, and 512 neurons.

Theorems & Definitions (20)

  • Proposition 1.1
  • Proposition 2.1: Local Estimates for $K$
  • proof
  • Lemma 2.2: ukai2007
  • Proposition 2.3: Local Estimates for $\Gamma$
  • proof
  • Lemma 3.1: hu2023higher
  • Theorem 3.2
  • proof
  • Remark 1
  • ...and 10 more