Convergence analysis of deep ReLU networks for solving integral equations

Abstract This paper proposes a deep rectified linear unit (ReLU) neural network framework for approximating solutions to linear and nonlinear integral equations, encompassing the Urysohn equation, Hammerstein equation, and weakly singular second-kind Fredholm integral equation. The proposed framework is trained through loss function minimization, exhibits significant superiority over conventional discretization-based approaches such as the Nyström method, collocation method, and Galerkin method. This is attributed to the mesh-free architecture, where samples for training the neural network are randomly drawn from the domain, enabling effective extension to high-dimensional integral equation problems. For the second-kind Fredholm integral equation, the convergence rate of this method is rigorously analyzed through the establishment of upper bounds on three critical parameters: the number of training samples, neurons per hidden layer, and network depth. The derived convergence rate comprises two distinct components: a statistical error term induced by Monte Carlo sampling, and an approximation error term stemming from ReLU network architecture constraints. Finally, numerical examples verify the validity of the proposed method.