揭示物理神经网络梯度传播失败根源，提出新架构有效解决该问题。
ProPINN: Demystifying Propagation Failures in Physics-Informed Neural Networks
- 发现PINN失败源于邻近采样点间梯度相关性低，而非传统认为的损失分布不均。
- 新架构ProPINN通过统一区域点梯度，使模型在复杂PDE求解中表现提升46%。
- 适合研究物理信息神经网络优化、数值方法与深度学习交叉领域的读者。
物理信息神经网络（PINNs）在求解偏微分方程（PDEs）方面备受期待，但其优化常因依赖导数的损失函数而面临严峻挑战。已有研究通过分析损失分布，观察到PINN的传播失败现象——即初始或边界处的正确监督无法有效传播至域内。本文超越直观理解，对传播失败进行形式化和深入研究。基于与经典有限元方法的对比，我们指出失败根源在于传统PINN的单点处理架构，并证明传播失败本质上由邻近采样点间梯度相关性较低所致。相较于表面损失图，这一新视角提供了更精确的定量判断标准，可定位失效位置与原因。该理论发现启发我们提出新型PINN架构ProPINN，通过联合区域点梯度实现更好传播。ProPINN能可靠克服原有失败模式，在多个测试中显著超越先进Transformer模型，相对性能提升达46%。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) have earned high expectations in solving partial differential equations (PDEs), but their optimization usually faces thorny challenges due to the unique derivative-dependent loss function. By analyzing the loss distribution, previous research observed the propagation failure phenomenon of PINNs, intuitively described as the correct supervision for model outputs cannot ''propagate'' from initial states or boundaries to the interior domain. Going beyond intuitive understanding, this paper provides a formal and in-depth study of propagation failure and its root cause. Based on a detailed comparison with classical finite element methods, we ascribe the failure to the conventional single-point-processing architecture of PINNs and further prove that propagation failure is essentially caused by the lower gradient correlation of PINN models on nearby collocation points. Compared to superficial loss maps, this new perspective provides a more precise quantitative criterion to identify where and why PINN fails. The theoretical finding also inspires us to present a new PINN architecture, named ProPINN, which can effectively unite the gradients of region points for better propagation. ProPINN can reliably resolve PINN failure modes and significantly surpass advanced Transformer-based models with 46% relative promotion.
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