用加权损失提升物理神经网络对玻尔兹曼方程的求解精度
A Theory-guided Weighted $L^2$ Loss for solving the BGK model via Physics-informed neural networks
- 引入速度加权的L2损失，重点惩罚高速区域误差
- 理论证明加权损失可保证解的收敛性，数值实验更准确稳定
- 适合需高精度模拟稀薄气体流动的研究者
物理信息神经网络为求解偏微分方程提供了有前景的框架，但将标准L2损失应用于玻尔兹曼-格罗斯-克鲁克（BGK）模型时存在根本缺陷。单纯最小化标准损失无法保证宏观量的准确预测，导致近似解无法捕捉真实物理解。为此，我们提出一种速度加权的L2损失函数，有效惩罚高速区域的误差。通过建立所提方法的稳定性估计，我们证明最小化该加权损失可保证近似解的收敛性。数值实验表明，采用此加权PINN损失在多个基准测试中均显著优于标准方法，具有更高的精度和鲁棒性。
原文摘要 · Abstract (English)
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.
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