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

DevFeed: [A Theory-guided Weighted $L^2$ Loss for solving the BGK model via Physics-informed neural networks](<https://devfeed.tech/articles/a-theory-guided-weighted-l-2-loss-for-solving-the-bgk-model-via-physics-informed-neural-networks-111389.md>)

Original publisher: [Read original article](<https://arxiv.org/abs/2604.04971>)

Author: Gyounghun Ko, Sung-Jun Son, Seung Yeon Cho, Myeong-Su Lee

Published: 2026-11-02T05:00:00Z

Content type: article

Language: en

Sources: [arXiv: Machine Learning](<https://devfeed.tech/sources/arxiv-machine-learning.md>)

Topics: [Neural Network](<https://devfeed.tech/topics/neural-network.md>), [Artificial Intelligence](<https://devfeed.tech/topics/ai.md>), [Finite Element Method (FEM)](<https://devfeed.tech/topics/finite-element-method.md>)

Tags: [convergence](<https://devfeed.tech/tags/convergence.md>), [cs-lg](<https://devfeed.tech/tags/cs-lg.md>), [cs-na](<https://devfeed.tech/tags/cs-na.md>), [errors](<https://devfeed.tech/tags/errors.md>), [experiments](<https://devfeed.tech/tags/experiments.md>), [math-na](<https://devfeed.tech/tags/math-na.md>), [model](<https://devfeed.tech/tags/model.md>), [neural-networks](<https://devfeed.tech/tags/neural-networks.md>), [partial-differential-equations](<https://devfeed.tech/tags/partial-differential-equations.md>), [physics](<https://devfeed.tech/tags/physics.md>), [physics-comp-ph](<https://devfeed.tech/tags/physics-comp-ph.md>), [physics-informed](<https://devfeed.tech/tags/physics-informed.md>), [robustness](<https://devfeed.tech/tags/robustness.md>), [solution](<https://devfeed.tech/tags/solution.md>), [standard](<https://devfeed.tech/tags/standard.md>), [theory](<https://devfeed.tech/tags/theory.md>), [velocity](<https://devfeed.tech/tags/velocity.md>)

## AI overview

For the BGK model, minimizing a standard PINN $L^2$ loss does not ensure accurate macroscopic moments or recovery of the physical solution. The authors propose a velocity-weighted $L^2$ loss that penalizes errors in high-velocity regions. They report a stability estimate under which minimizing this loss guarantees convergence of the approximate solution, and numerical experiments showing better accuracy and robustness than the standard approach across several benchmarks. The supplied text is an abstract and does not provide implementation details or benchmark specifics.

## Source excerpt

arXiv:2604.04971v2 Announce Type: replace 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.