面向 BGK 模型的理论引导加权 $L^2$ 损失：基于物理信息神经网络

虽然物理信息神经网络 (PINN) 为求解偏微分方程提供了有前景的框架，但标准 $L^2$ 损失函数在应用于巴坦纳-格罗斯-克鲁克 (BGK) 模型时是根本性不足的。 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.