Voronovskaya-Type Asymptotic Expansions and Convergence Analysis for Neural Network Operators in Complex Domains
Rômulo Damasclin Chaves dos Santos, Jorge Henrique de Oliveira Sales
TL;DR
The work develops Voronovskaya-type asymptotic expansions for generalized neural network operators defined on non-Euclidean and fractal domains, incorporating a modified activation $h_{q,\alpha}$ and density-based weights on manifolds. It introduces three operator families—$A_n$, $K_n$, and $Q_n$—built from a density $Z_{q,\alpha}$ and fractional derivatives to capture local and nonlocal behavior on curved spaces. The main results provide explicit expansions for $A_n$, establish fractional convergence for $Q_n$ with rate $O(1/n^{m-\beta})$, and prove uniform convergence on manifolds with negative curvature, supported by detailed proofs and quasi-interpolation analysis. These results offer a rigorous framework for function approximation, signal processing on manifolds, and solving PDEs on fractal geometries, with potential for numerical implementation and extension to dynamic systems.
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.