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.

翻译：本文研究了一类多元Kantorovich–Kernel型神经网络算子，包括Sharma与Singh所研究的深度Kantorovich型神经网络算子。我们证明了稠密性结果，建立了定量收敛估计，推导了Voronovskaya型定理，分析了深度复合算子偏微分方程的极限，证明了Korovkin型定理，并提出了反演定理。本文研究了一类多元Kantorovich–Kernel型神经网络算子，包括Sharma与Singh所研究的深度Kantorovich型神经网络算子。我们证明了稠密性结果，建立了定量收敛估计，推导了Voronovskaya型定理，分析了深度复合算子偏微分方程的极限，证明了Korovkin型定理，并提出了反演定理。此外，本文还讨论了神经网络架构与Chui、Hsu、He、Lorentz及Korovkin提出的经典正算子之间的联系。