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Revolutionizing Fractional Calculus with Neural Networks: Voronovskaya-Damasclin Theory for Next-Generation AI Systems

Rômulo Damasclin Chaves dos Santos, Jorge Henrique de Oliveira Sales

TL;DR

This work introduces rigorous convergence rates for neural network operators activated by symmetrized and perturbed hyperbolic tangent functions, utilizing novel Voronovskaya-Damasclin asymptotic expansions and extends classical approximation theory to fractional calculus via Caputo derivatives.

Abstract

This work introduces rigorous convergence rates for neural network operators activated by symmetrized and perturbed hyperbolic tangent functions, utilizing novel Voronovskaya-Damasclin asymptotic expansions. We analyze basic, Kantorovich, and quadrature-type operators over infinite domains, extending classical approximation theory to fractional calculus via Caputo derivatives. Key innovations include parameterized activation functions with asymmetry control, symmetrized density operators, and fractional Taylor expansions for error analysis. The main theorem demonstrates that Kantorovich operators achieve \(o(n^{-β(N-\varepsilon)})\) convergence rates, while basic operators exhibit \(\mathcal{O}(n^{-βN})\) error decay. For deep networks, we prove \(\mathcal{O}(L^{-β(N-\varepsilon)})\) approximation bounds. Stability results under parameter perturbations highlight operator robustness. By integrating neural approximation theory with fractional calculus, this work provides foundational mathematical insights and deployable engineering solutions, with potential applications in complex system modeling and signal processing.

Revolutionizing Fractional Calculus with Neural Networks: Voronovskaya-Damasclin Theory for Next-Generation AI Systems

TL;DR

This work introduces rigorous convergence rates for neural network operators activated by symmetrized and perturbed hyperbolic tangent functions, utilizing novel Voronovskaya-Damasclin asymptotic expansions and extends classical approximation theory to fractional calculus via Caputo derivatives.

Abstract

This work introduces rigorous convergence rates for neural network operators activated by symmetrized and perturbed hyperbolic tangent functions, utilizing novel Voronovskaya-Damasclin asymptotic expansions. We analyze basic, Kantorovich, and quadrature-type operators over infinite domains, extending classical approximation theory to fractional calculus via Caputo derivatives. Key innovations include parameterized activation functions with asymmetry control, symmetrized density operators, and fractional Taylor expansions for error analysis. The main theorem demonstrates that Kantorovich operators achieve \(o(n^{-β(N-\varepsilon)})\) convergence rates, while basic operators exhibit \(\mathcal{O}(n^{-βN})\) error decay. For deep networks, we prove \(\mathcal{O}(L^{-β(N-\varepsilon)})\) approximation bounds. Stability results under parameter perturbations highlight operator robustness. By integrating neural approximation theory with fractional calculus, this work provides foundational mathematical insights and deployable engineering solutions, with potential applications in complex system modeling and signal processing.

Paper Structure

This paper contains 19 sections, 8 theorems, 82 equations, 1 table.

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. Convergence of Operators in Deep Learning
  9. Fractional Perturbation Stability
  10. Generalized Voronovskaya Expansions
  11. Main Contribution:
  12. Error Term:
  13. Bounding the Remainder:
  14. Conclusion:
  15. Symmetrized Density Approach to Kantorovich Operator Convergence in Infinite Domains
  16. ...and 4 more sections

Key Result

Lemma 2.1

For all $x \in \mathbb{R}$ and $q, \lambda > 0$, the density function $M_{q, \lambda}(x)$ satisfies: This lemma ensures that $M_{q, \lambda}(x)$ is a valid probability density function, which is essential for the construction of neural network operators that approximate continuous functions.

Theorems & Definitions (17)

  • Definition 2.1: Perturbed Hyperbolic Tangent
  • Lemma 2.1: Positivity and Normalization of Density Function
  • proof
  • Theorem 3.1: Approximation by Operators
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3: Convergence of Operators in Deep Learning
  • proof
  • Theorem 4.1: Stability Under Fractional Perturbations
  • ...and 7 more