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

TL;DR

This paper introduces an advanced neural network operator framework that uses fractional and mixed activation functions to improve approximation accuracy in complex, high-dimensional spaces, supported by theoretical and numerical validation.

Contribution

It extends symmetrized neural network operators by incorporating fractional and mixed activation functions, enhancing their ability to approximate higher-order smooth functions.

Findings

Improved approximation accuracy for complex functions.

Established uniform convergence rates with new inequalities.

Validated efficiency through numerical experiments.

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 $\theta$-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.