A Theory-guided Weighted $L^2$ Loss for solving the BGK model via Physics-informed neural networks

TL;DR

This paper introduces a velocity-weighted $L^2$ loss for physics-informed neural networks to improve the accuracy of solutions to the BGK model, ensuring better convergence and physical fidelity.

Contribution

The authors propose a novel weighted loss function tailored for the BGK model, with theoretical stability guarantees and improved numerical performance over standard methods.

Findings

Weighted loss enhances accuracy in high-velocity regions.

Theoretical stability guarantees convergence of the solution.

Numerical experiments show superior robustness and accuracy.

Abstract

While Physics-Informed Neural Networks offer a promising framework for solving partial differential equations, the standard $L^2$ loss formulation is fundamentally insufficient when applied to the Bhatnagar-Gross-Krook (BGK) model. Specifically, simply minimizing the standard loss does not guarantee accurate predictions of the macroscopic moments, causing the approximate solutions to fail in capturing the true physical solution. To overcome this limitation, we introduce a velocity-weighted $L^2$ loss function designed to effectively penalize errors in the high-velocity regions. By establishing a stability estimate for the proposed approach, we shows that minimizing the proposed weighted loss guarantees the convergence of the approximate solution. Also, numerical experiments demonstrate that employing this weighted PINN loss leads to superior accuracy and robustness across various benchmarks compared to the standard approach.