Generalized Neural Network Operators with Symmetrized Activations: Fractional Convergence and the Voronovskaya-Damasclin Theorem

TL;DR

This paper investigates the asymptotic behavior and convergence properties of neural network operators with symmetrized activations, extending classical approximation results to fractional calculus and establishing new theorems for error estimation.

Contribution

It introduces a novel Voronovskaya-Damasclin theorem for neural network operators with symmetrized activations, incorporating fractional derivatives and providing precise convergence analysis.

Findings

Derived Voronovskaya-type expansions for neural network operators.

Extended classical approximation results to fractional calculus using Caputo derivatives.

Established convergence rates and error estimates for the operators.

Abstract

This paper explores the asymptotic behavior of univariate neural network operators, with an emphasis on both classical and fractional differentiation over infinite domains. The analysis leverages symmetrized and perturbed hyperbolic tangent activation functions to investigate basic, Kantorovich, and quadrature-type operators. Voronovskaya-type expansions, along with the novel Vonorovskaya-Damasclin theorem, are derived to obtain precise error estimates and establish convergence rates, thereby extending classical results to fractional calculus via Caputo derivatives. The study delves into the intricate interplay between operator parameters and approximation accuracy, providing a comprehensive framework for future research in multidimensional and stochastic settings. This work lays the groundwork for a deeper understanding of neural network operators in complex mathematical.