Kantorovich--Kernel Neural Operators: Approximation Theory, Asymptotics, and Neural Network Interpretation

TL;DR

This paper explores Kantorovich-kernel neural operators, providing theoretical convergence results, limit analyses, and connections to classical positive operators, advancing the mathematical understanding of neural network approximation.

Contribution

It introduces new approximation and convergence theorems for multivariate Kantorovich-kernel neural operators, linking neural architectures with classical operator theory.

Findings

Proved density and convergence estimates for the operators.

Derived Voronovskaya-type theorems and analyzed PDE limits.

Established connections between neural networks and classical positive operators.

Abstract

This paper studies a class of multivariate Kantorovich-kernel neural network operators, including the deep Kantorovich-type neural network operators studied by Sharma and Singh. We prove density results, establish quantitative convergence estimates, derive Voronovskaya-type theorems, analyze the limits of partial differential equations for deep composite operators, prove Korovkin-type theorems, and propose inversion theorems. This paper studies a class of multivariate Kantorovich-kernel neural network operators, including the deep Kantorovich-type neural network operators studied by Sharma and Singh. We prove density results, establish quantitative convergence estimates, derive Voronovskaya-type theorems, analyze the limits of partial differential equations for deep composite operators, prove Korovkin-type theorems, and propose inversion theorems. Furthermore, this paper discusses the connection between neural network architectures and the classical positive operators proposed by Chui, Hsu, He, Lorentz, and Korovkin.