提出变量分解策略，解决PINNs因导数失控导致的求解失效问题。
Beyond Derivative Pathology of PINNs: Variable Splitting Strategy with Convergence Analysis
- 将解的梯度设为辅助变量，直接调控导数行为。
- 证明该方法对二阶线性PDE可收敛到广义解。
- 适合需要高精度求解PDE的科研与工程场景。
物理信息神经网络（PINNs）近年来被广泛用于求解各类偏微分方程（PDE）。现有研究多关注其预测不准的失败模式，但普遍基于一个前提：损失函数趋近零即意味着网络收敛到真实PDE解。本文证明该前提根本无效，问题根源在于无法控制预测解的导数行为。受此“导数病理”启发，我们提出“变量分解”策略，将解的梯度作为辅助变量进行参数化。该方法通过直接监控和调节梯度，有效规避导数病理。进一步，我们证明所提方法对二阶线性PDE能保证收敛至广义解，表明其具有广泛的适用性。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) have recently emerged as effective methods for solving partial differential equations (PDEs) in various problems. Substantial research focuses on the failure modes of PINNs due to their frequent inaccuracies in predictions. However, most are based on the premise that minimizing the loss function to zero causes the network to converge to a solution of the governing PDE. In this study, we prove that PINNs encounter a fundamental issue that the premise is invalid. We also reveal that this issue stems from the inability to regulate the behavior of the derivatives of the predicted solution. Inspired by the \textit{derivative pathology} of PINNs, we propose a \textit{variable splitting} strategy that addresses this issue by parameterizing the gradient of the solution as an auxiliary variable. We demonstrate that using the auxiliary variable eludes derivative pathology by enabling direct monitoring and regulation of the gradient of the predicted solution. Moreover, we prove that the proposed method guarantees convergence to a generalized solution for second-order linear PDEs, indicating its applicability to various problems.
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