Error Estimation for Physics-informed Neural Networks Approximating Semilinear Wave Equations

Understand

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.