Physics-informed neural networks (PINNs) have shown promise in solving various partial differential equations (PDEs). However, training pathologies have negatively affected the convergence and prediction accuracy of PINNs, which further limits their practical applications. In this paper, we propose to use condition number as a metric to diagnose and mitigate the pathologies in PINNs. Inspired by classical numerical analysis, where the condition number measures sensitivity and stability, we highlight its pivotal role in the training dynamics of PINNs. We prove theorems to reveal how condition number is related to both the error control and convergence of PINNs. Subsequently, we present an algorithm that leverages preconditioning to improve the condition number. Evaluations of 18 PDE problems showcase the superior performance of our method. Significantly, in 7 of these problems, our method reduces errors by an order of magnitude. These empirical findings verify the critical role of the condition number in PINNs' training.

翻译：物理信息神经网络（PINNs）在求解各类偏微分方程（PDEs）方面展现出潜力。然而，训练病理现象对PINNs的收敛性和预测精度造成了负面影响，进而限制了其实际应用。本文提出以条件数作为指标来诊断和缓解PINNs中的病理问题。受经典数值分析中条件数衡量敏感性与稳定性的启发，我们揭示了条件数在PINNs训练动态中的关键作用。通过定理证明，阐明了条件数与PINNs误差控制和收敛性之间的内在关联。进而提出一种利用预处理改善条件数的算法。在18个偏微分方程问题上的评估表明，本方法具有优越性能。特别在7个问题中，该方法将误差降低了一个数量级。这些实验结果验证了条件数在PINNs训练中的关键作用。