Kantorovich--Kernel Neural Operators: Approximation Theory, Asymptotics, and Neural Network Interpretation
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908917024423936 |
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| author | He, Tian-Xiao |
| author_facet | He, Tian-Xiao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26418 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kantorovich--Kernel Neural Operators: Approximation Theory, Asymptotics, and Neural Network Interpretation He, Tian-Xiao Machine Learning Functional Analysis 41A36, 41A25, 47D07, 35K55 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. |
| title | Kantorovich--Kernel Neural Operators: Approximation Theory, Asymptotics, and Neural Network Interpretation |
| topic | Machine Learning Functional Analysis 41A36, 41A25, 47D07, 35K55 |
| url | https://arxiv.org/abs/2603.26418 |