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

Rômulo Damasclin Chaves dos Santos

TL;DR

The paper develops Voronovskaya-type asymptotics for neural network operators that use symmetrized and perturbed hyperbolic tangent activations to approximate both classical and fractional derivatives on infinite domains. By introducing a symmetrized density $\Phi$ and employing Caputo fractional derivatives, the authors derive precise error expansions for basic, Kantorovich, and quadrature-type operators, including stability under small perturbations of the activation and generalized density. The main contributions include the Voronovskaya-Damasclin theorem and its fractional-extension, providing explicit remainder bounds of the form $o(n^{-\beta(N-\varepsilon)})$, and a framework that paves the way for stochastic and multivariate generalizations. The results offer rigorous guidance for designing neural operators with provable convergence properties in fractional-calculus contexts, with potential impact on fluid dynamics, signal processing, and numerical analysis of fractional differential equations.

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.

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

TL;DR

The paper develops Voronovskaya-type asymptotics for neural network operators that use symmetrized and perturbed hyperbolic tangent activations to approximate both classical and fractional derivatives on infinite domains. By introducing a symmetrized density and employing Caputo fractional derivatives, the authors derive precise error expansions for basic, Kantorovich, and quadrature-type operators, including stability under small perturbations of the activation and generalized density. The main contributions include the Voronovskaya-Damasclin theorem and its fractional-extension, providing explicit remainder bounds of the form , and a framework that paves the way for stochastic and multivariate generalizations. The results offer rigorous guidance for designing neural operators with provable convergence properties in fractional-calculus contexts, with potential impact on fluid dynamics, signal processing, and numerical analysis of fractional differential equations.

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.

Paper Structure

This paper contains 37 sections, 6 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Mathematical Foundations of Symmetrized Activation Functions
  3. Main Results
  4. Basic Operators
  5. Case 1: $\left| \frac{k}{n} - x \right| < \frac{1}{n^\beta}$.
  6. Case 2: $\left| \frac{k}{n} - x \right| \geq \frac{1}{n^\beta}$.
  7. Kantorovich Operators
  8. Fractional Perturbation Stability
  9. Generalized Voronovskaya Expansions for Fractional Functions
  10. Main Contribution:
  11. Error Term:
  12. Bounding the Remainder:
  13. Conclusion:
  14. Convergence of Kantorovich Operators with Generalized Density
  15. Main Contribution:
  16. ...and 22 more sections

Key Result

Theorem 1

Let $0 < \beta < 1$, $n \in \mathbb{N}$ be sufficiently large, $x \in \mathbb{R}$, $f \in C^N(\mathbb{R})$ such that $f^{(N)} \in C_B(\mathbb{R})$ (bounded and continuous), and $0 < \varepsilon \leq N$. Then:

Theorems & Definitions (12)

  • Theorem 1: Approximation by Operators
  • proof
  • Theorem 2
  • proof
  • Theorem 3: Stability Under Fractional Perturbations
  • proof
  • Theorem 4: Generalized Voronovskaya Expansion
  • proof
  • Theorem 5: Convergence Under Generalized Density
  • proof
  • ...and 2 more