Max-Min Neural Network Operators For Approximation of Multivariate Functions

TL;DR

This paper introduces a multivariate max-min neural network operator framework that extends univariate approximation techniques, providing convergence theorems and quantitative estimates for stable and efficient multivariate function approximation.

Contribution

It develops and analyzes new multivariate max-min neural network operators activated by sigmoidal functions, advancing approximation theory with proven convergence and error estimates.

Findings

Established pointwise and uniform convergence theorems.

Derived quantitative approximation estimates using modulus of continuity.

Demonstrated the efficiency and stability of the multivariate max-min operators.

Abstract

In this paper, we develop a multivariate framework for approximation by max-min neural network operators. Building on the recent advances in approximation theory by neural network operators, particularly, the univariate max-min operators, we propose and analyze new multivariate operators activated by sigmoidal functions. We establish pointwise and uniform convergence theorems and derive quantitative estimates for the order of approximation via modulus of continuity and multivariate generalized absolute moment. Our results demonstrate that multivariate max-min structure of operators, besides their algebraic elegance, provide efficient and stable approximation tools in both theoretical and applied settings.