Physics-informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of partial differential equations (PDEs). We provide 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.

中文翻译：

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估计物理信息神经网络用于逼近 PDE 的泛化误差

物理信息神经网络 (PINN) 最近已广泛用于偏微分方程 (PDE) 的稳健和准确逼近。我们提供了 PINN 的泛化误差的上限，近似于 PDE 的前向问题的解决方案。引入了一种抽象的形式，并利用基础 PDE 的稳定性属性来推导根据训练误差和训练样本数量对泛化误差的估计。这个抽象框架用几个非线性偏微分方程的例子来说明。还提出了验证所提出理论的数值实验。