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

TL;DR

This work develops a class of symmetry-preserving neural network operators for multivariate function approximation by employing a parameterized activation $g_{q,\lambda}$ and a symmetric density-based framework. It extends Voronovskaya-type expansions to incorporate higher-order interactions via operator families $A_n$, $K_n$, and $Q_n$, and establishes convergence and error bounds in Sobolev spaces ($W^{m,p}$) and Hölder spaces ($C^{m,\alpha}$). The key theoretical contributions are Theorem 1 (symmetry-preserving convergence for $A_n$), Theorem 2 (Sobolev-space error bounds for $K_n$ and $Q_n$), and Theorem 3 (Hölder-space convergence for all three operators), underpinned by a multivariate density $Z(x) = \prod_{i=1}^N \Phi(x_i)$ built from $\Phi$. The results provide a rigorous foundation for symmetry-driven modeling and regularization in machine learning, with potential applications in multivariate interpolation, image reconstruction, and time-series analysis that require preserving dynamic symmetries and higher-order interactions. All mathematical notation is employed to articulate the operators, spaces, and convergence rates essential for theoretical and applied adoption.

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.