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Extension of Symmetrized Neural Network Operators with Fractional and Mixed Activation Functions

Rômulo Damasclin Chaves dos Santos, Jorge Henrique de Oliveira Sales

TL;DR

The paper extends symmetrized neural network operators by integrating fractional and mixed activation functions to better approximate higher-order smooth functions in complex, high-dimensional domains. It defines a $q$-deformed, $\\theta$-parametrized density via $\\phi_{q,\\theta,\\alpha}(x)=\\frac{1}{1+q^{A\\theta|x|^{\\alpha}}}$ and a corresponding density $W_{q,\\theta,\\alpha}(x)$, yielding the fractional operator $S_n(f; x)=\\sum_{k=-\\infty}^{\\infty} f(\\frac{k}{n}) W_{q,\\theta,\\alpha}(n x - k)$. Theoretical contributions include Jackson-type inequalities $\\|S_n(f; x) - f(x)\\| \\le C \\omega_2\left(f, \\frac{1}{n}\\right)$, uniform convergence, convergence-rate bounds, and operator stability within $L^p$, Hölder, and Sobolev spaces, supported by numerical validations. These results broaden neural network approximation theory to broader functional spaces, enabling applications to PDEs and modeling of complex systems with oscillatory or fractional components.

Abstract

We propose a novel extension to symmetrized neural network operators by incorporating fractional and mixed activation functions. This study addresses the limitations of existing models in approximating higher-order smooth functions, particularly in complex and high-dimensional spaces. Our framework introduces a fractional exponent in the activation functions, allowing adaptive non-linear approximations with improved accuracy. We define new density functions based on $q$-deformed and $θ$-parametrized logistic models and derive advanced Jackson-type inequalities that establish uniform convergence rates. Additionally, we provide a rigorous mathematical foundation for the proposed operators, supported by numerical validations demonstrating their efficiency in handling oscillatory and fractional components. The results extend the applicability of neural network approximation theory to broader functional spaces, paving the way for applications in solving partial differential equations and modeling complex systems.

Extension of Symmetrized Neural Network Operators with Fractional and Mixed Activation Functions

TL;DR

The paper extends symmetrized neural network operators by integrating fractional and mixed activation functions to better approximate higher-order smooth functions in complex, high-dimensional domains. It defines a -deformed, -parametrized density via and a corresponding density , yielding the fractional operator . Theoretical contributions include Jackson-type inequalities , uniform convergence, convergence-rate bounds, and operator stability within , Hölder, and Sobolev spaces, supported by numerical validations. These results broaden neural network approximation theory to broader functional spaces, enabling applications to PDEs and modeling of complex systems with oscillatory or fractional components.

Abstract

We propose a novel extension to symmetrized neural network operators by incorporating fractional and mixed activation functions. This study addresses the limitations of existing models in approximating higher-order smooth functions, particularly in complex and high-dimensional spaces. Our framework introduces a fractional exponent in the activation functions, allowing adaptive non-linear approximations with improved accuracy. We define new density functions based on -deformed and -parametrized logistic models and derive advanced Jackson-type inequalities that establish uniform convergence rates. Additionally, we provide a rigorous mathematical foundation for the proposed operators, supported by numerical validations demonstrating their efficiency in handling oscillatory and fractional components. The results extend the applicability of neural network approximation theory to broader functional spaces, paving the way for applications in solving partial differential equations and modeling complex systems.
Paper Structure (20 sections, 4 theorems, 73 equations, 1 table)

This paper contains 20 sections, 4 theorems, 73 equations, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Main Results
  4. Jackson-Type Inequalities
  5. Uniform Convergence
  6. Convergence Rate
  7. Stability of the Operators
  8. Results
  9. Conclusions
  10. Additional Details on Fractional Activation Functions
  11. Definition and Properties
  12. Symmetrized Density Function
  13. Moments of the Density Function
  14. Function Spaces and Norms
  15. Hölder Spaces
  16. ...and 5 more sections

Key Result

Theorem 3.1

Let $f \in C^2([-a, a], \mathbb{C})$. Then, for $n \in \mathbb{N}$ and $x \in [-a, a]$: where $\omega_2(f, t)$ is the second-order modulus of continuity.

Theorems & Definitions (10)

  • Theorem 3.1: Jackson Inequality for Fractional Operators
  • proof
  • Theorem 3.2: Uniform Convergence
  • proof
  • Theorem 3.3: Convergence Rate
  • proof
  • Theorem 3.4: Stability of Fractional Symmetrized Neural Network Operators
  • proof
  • proof
  • proof