Simultaneous approximation by neural network operators with applications to Voronovskaja formulas

TL;DR

This paper investigates the simultaneous approximation of functions and their derivatives using neural network operators with sigmoidal activation, providing convergence theorems, quantitative estimates, and Voronovskaja-type formulas.

Contribution

It introduces new uniform convergence and approximation order results for neural network operators, including high-order approximation insights via Voronovskaja formulas.

Findings

Established uniform convergence for derivatives of NN operators

Provided quantitative approximation estimates based on modulus of continuity

Derived Voronovskaja-type formulas for high-order approximation

Abstract

In this paper, we considered the problem of the simultaneous approximation of a function and its derivatives by means of the well-known neural network (NN) operators activated by sigmoidal function. Other than a uniform convergence theorem for the derivatives of NN operators, we also provide a quantitative estimate for the order of approximation based on the modulus of continuity of the approximated derivative. Furthermore, a qualitative and quantitative Voronovskaja-type formula is established, which provides information about the high order of approximation that can be achieved by NN operators. To prove the above theorems, several auxiliary results involving sigmoidal functions have been established. At the end of the paper, noteworthy examples have been discussed in detail.