Univariate error function based neural network approximation

TL;DR

This paper investigates neural network operators based on the error function for approximating continuous functions on intervals and the real line, providing theoretical bounds and fractional approximation methods.

Contribution

It introduces a new class of neural network operators using the error function, establishing Jackson type inequalities and extending to fractional approximation.

Findings

Derived pointwise and uniform approximation bounds.

Established Jackson type inequalities involving modulus of continuity.

Extended approximation methods to fractional derivatives.

Abstract

Here we research the univariate quantitative approximation of real and complex valued continuous functions on a compact interval or all the real line by quasi-interpolation, Baskakov type and quadrature type neural network operators. We perform also the related fractional approximation. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its high order derivative or fractional derivatives. Our operators are defined by using a density function induced by the error function. The approximations are pointwise and with respect to the uniform norm. The related feed-forward neural networks are with one hidden layer.