Error Estimation for Physics-informed Neural Networks Approximating Semilinear Wave Equations
This work addresses the need for theoretical guarantees in physics-informed neural networks for researchers in computational physics and machine learning, but it is incremental as it focuses on specific equations and network architectures.
The paper tackles the problem of providing rigorous error bounds for physics-informed neural networks approximating semilinear wave equations, resulting in a bound for the total error in a specific norm that can be made arbitrarily small under certain assumptions, with numerical experiments illustrating the theoretical bounds.
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.