This paper provides rigorous error bounds for physics-informed neural networks approximating the semilinear wave equation. We provide bounds for the generalization and training error in terms of the width of the network's layers and the number of training points for a tanh neural network with two hidden layers. Our main result is a bound of the total error in the $H^1([0,T];L^2(Ω))$-norm in terms of the training error and the number of training points, which can be made arbitrarily small under some assumptions. We illustrate our theoretical bounds with numerical experiments.

翻译：本文为逼近半线性波动方程的物理信息神经网络提供了严格的误差界。针对具有两个隐藏层的tanh神经网络，我们给出了泛化误差和训练误差关于网络层宽度和训练点数量的界。主要结果是在$H^1([0,T];L^2(Ω))$范数下，总误差由训练误差和训练点数量决定，且在某些假设下该误差可任意小。我们通过数值实验验证了理论界的有效性。