Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer

Abstract

Let σ be a non-polynomial activation function of a neural network that has nth order continuous derivatives on R. The objective of this paper is to show that for any compact set K of R s , s ≥ 1, and any multivariate function f defined on an open set containing K, a neural network with one hidden layer can be so constructed that f and all its existing continuous kth order partial derivatives, for k = ( k 1, …, k s ) ϵ Z + s satisfying ∑ i = 1 s k i ≤ n, can be simultaneously and uniformly approximated on K by the network.