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

Solving these equations analytically is often challenging or even impossible, necessitating the utilization of other methods to obtain approximate solutions. One way to find approximate solutions to partial differential equations is through classical numerical methods. These methods have been studied for years and already have strong theoretical foundations when it comes to error estimation [1]. However, in recent years, with the rise of machine learning as a whole, there has also been an increased interest in applying machine learning methods to the problem of finding approximate solutions to PDEs. As universal function approximators [2], deep neural networks provide a promising avenue for a multitude of approaches to the approximation of solutions to partial differential equations. Among these methods are neural operators, methods based on the Feynman-Kac formula, and methods for parametric PDEs [3] [4] [5]. This paper focuses on physics-informed neural networks (PINNs), which were conceived as feed-forward neural networks that incorporate the dynamics of the PDE into their loss function [6].