提出自适应加权方法，提升物理信息神经网络求解精度
Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective
- 从点级损失转向邻域连续加权，优化训练分布
- 相对L2误差显著降低，收敛性与精度得到改善
- 适合求解偏微分方程的科研人员参考
物理信息神经网络（PINNs）广泛用于求解偏微分方程（PDEs），通过约束深度学习模型的输出和梯度满足控制方程。然而，受限于计算能力，PINNs通常在有限采样点上进行优化，这给收敛性和精度带来了严峻挑战。本文提出一种新的加权方案，可自适应地将损失函数的权重从孤立点扩展至其连续邻域区域。实验结果表明，该加权方案能有效降低相对$L^2$误差至更低水平。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs) by ensuring that the outputs and gradients of deep learning models adhere to the governing equations. However, constrained by computational limitations, PINNs are typically optimized using a finite set of points, which poses significant challenges in guaranteeing their convergence and accuracy. In this study, we proposed a new weighting scheme that will adaptively change the weights to the loss functions from isolated points to their continuous neighborhood regions. The empirical results show that our weighting scheme can reduce the relative $L^2$ errors to a lower value.
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