Voronovskaya-Type Asymptotic Expansions and Convergence Analysis for Neural Network Operators in Complex Domains

TL;DR

This paper extends Voronovskaya-type asymptotic expansions to neural network operators on complex, non-Euclidean, and fractal domains, analyzing their convergence and applicability to structured spaces.

Contribution

It introduces generalized neural network operators on complex geometries, incorporating fractional derivatives and asymptotic analysis, advancing the mathematical understanding of neural networks in complex domains.

Findings

Operators preserve density properties.

Established convergence rates and asymptotic expansions.

Demonstrated applicability to fractal and manifold geometries.

Abstract

This paper extends the classical theory of Voronovskaya-type asymptotic expansions to generalized neural network operators defined on non-Euclidean and fractal domains. We introduce and analyze smooth operators activated by modified and generalized hyperbolic tangent functions, extending their applicability to manifold and fractal geometries. Key theoretical results include the preservation of density properties, detailed convergence rates, and asymptotic expansions. Additionally, we explore the role of fractional derivatives in defining neural network operators, which capture non-local behavior and are particularly useful for modeling systems with long-range dependencies or fractal-like structures. Our findings contribute to a deeper understanding of neural network operators in complex, structured spaces, offering robust mathematical tools for applications in signal processing on manifolds and solving partial differential equations on fractals.