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

Tian-Xiao He

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arXiv · 2603.26418

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

Tian-Xiao He

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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.

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Tian-Xiao He. 2026-03-27. Kantorovich--Kernel Neural Operators: Approximation Theory, Asymptotics, and Neural Network Interpretation. https://arxiv.org/abs/2603.26418

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