A Priori Estimation of the Approximation, Optimization and Generalization Error of Random Neural Networks for Solving Partial Differential Equations

In recent years, there are numerous methods involving neural networks for solving partial differential equations (PDEs), such as Physics informed neural networks (PINNs), Deep Ritz method (DRM) and others. However, the optimization problems are typically non-convex, which makes these methods lead to unsatisfactory solutions. With weights sampled from some distribution, applying random neural networks to solve PDEs yields least squares problems that are easily solvable. In this paper, we focus on Barron type functions and demonstrate the approximation, optimization and generalization of random neural networks for solving PDEs.