Algebraic function based Banach space valued ordinary and fractional neural network approximations

TL;DR

This paper investigates the approximation of Banach space valued functions using neural networks with algebraic sigmoid functions, establishing Jackson type inequalities for both ordinary and fractional cases.

Contribution

It introduces a new class of neural network operators for Banach space valued functions and derives approximation bounds involving fractional derivatives.

Findings

Established Jackson type inequalities for neural network approximations.

Demonstrated pointwise and uniform norm approximation capabilities.

Utilized algebraic sigmoid functions in neural network operator design.

Abstract

Here we research the univariate quantitative approximation, ordinary and fractional, of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative of fractional derivatives. Our operators are defined by using a density function generated by an algebraic sigmoid function. The approximations are pointwise and of the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer.