Non-Linear Interactions in Neural Network Operators: New Theorems on Symmetry-Preserving Transformations

TL;DR

This paper introduces new symmetry-preserving neural network operators using hyperbolic tangent functions, providing theoretical guarantees for their convergence and improved approximation capabilities in multivariate settings.

Contribution

It presents novel non-linear operators that extend symmetry-preserving transformations with rigorous convergence theorems and error bounds.

Findings

Operators achieve higher accuracy in function approximation.

Theoretical convergence and error bounds are established.

Operators effectively model higher-order interactions in multivariate data.

Abstract

This paper advances the study of multivariate function approximation using neural network operators activated by symmetrized and perturbed hyperbolic tangent functions. We propose new non-linear operators that preserve dynamic symmetries within Euclidean spaces, extending current results on Voronovskaya-type asymptotic expansions. The developed operators utilize parameterized deformations to model higher-order interactions in multivariate settings, achieving improved accuracy and robustness. Fundamental theorems demonstrate convergence properties and quantify error bounds, highlighting the operators' ability to approximate functions and derivatives in Sobolev spaces. These results provide a rigorous foundation for theoretical studies and further applications in symmetry-driven modeling and regularization strategies in machine learning and data analysis.