标题： 物理信息神经网络的预处理 显示英文标题

标题： Preconditioning for Physics-Informed Neural Networks

摘要： 物理信息神经网络（PINNs）在解决各种偏微分方程（PDEs）方面表现出潜力。 然而，训练病态对PINNs的收敛性和预测准确性产生了负面影响，这进一步限制了它们的实际应用。 在本文中，我们提出使用条件数作为诊断和缓解PINNs中病态的度量。 受经典数值分析的启发，在经典数值分析中，条件数衡量敏感性和稳定性，我们强调其在PINNs训练动态中的关键作用。 我们证明了定理，以揭示条件数如何与PINNs的误差控制和收敛性相关。 随后，我们提出了一种利用预处理来改善条件数的算法。 对18个PDE问题的评估展示了我们方法的优越性能。 显著的是，在这些问题中的7个问题中，我们的方法将误差降低了数量级。 这些实证结果验证了条件数在PINNs训练中的关键作用。 显示英文摘要

摘要： 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.