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 $H^1([0,T];L^2(\Omega))$-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：本文为物理信息神经网络逼近半线性波动方程提供了严格的误差界。我们针对具有两个隐藏层的tanh神经网络，给出了其泛化误差和训练误差关于网络层宽度及训练点数量的界。主要结果给出了总误差在$H^1([0,T];L^2(\Omega))$范数下关于训练误差和训练点数量的界，该误差在若干假设下可被任意缩小。我们还通过数值实验验证了理论界。