A Theory-guided Weighted $L^2$ Loss for solving the BGK model via Physics-informed neural networks
For researchers using PINNs to solve kinetic equations, this work addresses a fundamental failure mode of standard loss functions.
The paper identifies that standard L2 loss in PINNs fails for the BGK model, and proposes a velocity-weighted L2 loss that guarantees convergence of macroscopic moments. Numerical experiments show superior accuracy and robustness over the standard approach.
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.