General multivariate arctangent function activated neural network approximations

Abstract

Here we expose multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or \(\mathbb{R}^{N}\), \(N\in \mathbb{N}\), by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Frechet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the arctangent function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.