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Error estimates of asymptotic-preserving neural networks in approximating stochastic linearized Boltzmann equation

Jiayu Wan, Liu Liu

TL;DR

This paper constructs an asymptotic-preserving neural networks (APNNs) for the linearized Boltzmann equation in the acoustic scaling and with uncertain parameters and demonstrates the existence of APNNs when the loss function approaches zero, and the convergence of the APNN approximated solution as the loss tends to zero.

Abstract

In this paper, we construct an asymptotic-preserving neural networks (APNNs) [21] for the linearized Boltzmann equation in the acoustic scaling and with uncertain parameters. Utilizing the micro-macro decomposition, we design the loss function based on the stochastic-Galerkin system conducted from the micro-macro equations. Rigorous analysis is provided to show the capability of neural networks in approximating solutions near the global Maxwellian. By employing hypocoercivity techniques, we demonstrate two key results: the existence of APNNs when the loss function approaches zero, and the convergence of the APNN approximated solution as the loss tends to zero, with the error exhibiting an exponential decay in time.

Error estimates of asymptotic-preserving neural networks in approximating stochastic linearized Boltzmann equation

TL;DR

This paper constructs an asymptotic-preserving neural networks (APNNs) for the linearized Boltzmann equation in the acoustic scaling and with uncertain parameters and demonstrates the existence of APNNs when the loss function approaches zero, and the convergence of the APNN approximated solution as the loss tends to zero.

Abstract

In this paper, we construct an asymptotic-preserving neural networks (APNNs) [21] for the linearized Boltzmann equation in the acoustic scaling and with uncertain parameters. Utilizing the micro-macro decomposition, we design the loss function based on the stochastic-Galerkin system conducted from the micro-macro equations. Rigorous analysis is provided to show the capability of neural networks in approximating solutions near the global Maxwellian. By employing hypocoercivity techniques, we demonstrate two key results: the existence of APNNs when the loss function approaches zero, and the convergence of the APNN approximated solution as the loss tends to zero, with the error exhibiting an exponential decay in time.

Paper Structure

This paper contains 11 sections, 8 theorems, 144 equations.

Table of Contents

  1. Preliminaries
  2. Introduction to Boltzmann equation with uncertainties
  3. Hypocoercivity assumptions.
  4. Stochastic Galerkin (SG) method
  5. APNN framework
  6. Micro-macro decomposition
  7. APNN for the SG system
  8. Main results
  9. Existence of APNN with arbitrarily small loss
  10. Convergence of APNN solution
  11. Conclusion

Key Result

Theorem 1

Consider the linearized Boltzmann equation with uncertainty fully linear Botzmann with UQ, and let $(\bm{\rho},\bm{u},\bm{\theta},\bm{g})$ be the solution to the macro-micro formulation MM_formulation_SG such that the microscopic part $\bm{g}$ satisfies Assumption 1. Then for any $\delta>0$, there e

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 2
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Remark 1