# A Theory-guided Weighted L2 Loss for solving the BGK model via Physics-informed neural networks

arXiv: https://arxiv.org/abs/2604.04971 · HTML: https://mlsanity.com/paper/2604.04971/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 41 references that are in the map.
- First posted on arXiv 4 Apr 2026.

## Builds on (most cited first, 9 of 9)

- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach](https://mlsanity.com/paper/1911.09843/) (2019)
- [opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation](https://mlsanity.com/paper/2207.01765/) (2022)
- [Separable Physics-informed Neural Networks for Solving the BGK Model of the Boltzmann Equation](https://mlsanity.com/paper/2403.06342/) (2024)
- [Semi-analytic PINN methods for singularly perturbed boundary value problems](https://mlsanity.com/paper/2208.09145/) (2022)
- [Forward and Inverse Simulation of Pseudo-Two-Dimensional Model of Lithium-Ion Batteries Using Neural Networks](https://mlsanity.com/paper/2412.13200/) (2024)

## Abstract (arXiv)

While Physics-Informed Neural Networks offer a promising framework for solving partial differential equations, the standard L2 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 L2 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.