Universal approximation property of a continuous neural network based on a nonlinear diffusion equation

Abstract

Recently, differential equation-based neural networks have been actively studied. This paper discusses the universal approximation property of a neural network that is based on a nonlinear partial differential equation (PDE) of the parabolic type.Based on the assumption that the activation function is non-polynomial and Lipschitz continuous, and applying the theory of the difference method, we show that an arbitrary continuous function on any compact set can be approximated using the output of the network with arbitrary precision. Additionally, we present an estimate of the order of accuracy with respect to △t and △x.