提出新型神经网络算子，解析其逼近理论与微分方程极限行为。
Kantorovich--Kernel Neural Operators: Approximation Theory, Asymptotics, and Neural Network Interpretation
- 基于Kantorovich核构建多变量神经网络算子，统一分析架构
- 给出收敛速率、渐近展开及深度复合算子的微分方程极限
- 揭示神经网络与经典正算子的数学联系，适合理论研究者
本文研究一类多变量Kantorovich-核神经网络算子，涵盖Sharma和Singh研究的深层Kantorovich型神经网络算子。证明了稠密性结果，建立了定量收敛估计，推导了Voronovskaya型定理，分析了深层复合算子的偏微分方程极限，证明了Korovkin型定理，并提出了反演定理。此外，本文探讨了神经网络架构与Chui、Hsu、He、Lorentz和Korovkin提出的经典正算子之间的联系。
原文摘要 · Abstract (English)
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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