Physics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of training samples. This abstract framework is illustrated with several examples of nonlinear PDEs. Numerical experiments, validating the proposed theory, are also presented.

翻译：物理信息神经网络（PINNs）近年来被广泛应用于偏微分方程的鲁棒且精确逼近。本文针对PINNs逼近偏微分方程正问题解的过程，给出了泛化误差的严格上界。我们引入了一种抽象形式体系，并利用底层PDE的稳定性性质，推导出以训练误差和训练样本数量表示的泛化误差估计。该抽象框架通过多个非线性PDE实例加以说明。此外，本文还提供了验证所提出理论的数值实验。