Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911719467515904 |
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| author | Chen, Shuang He, Juncai Tai, Xue-Cheng |
| author_facet | Chen, Shuang He, Juncai Tai, Xue-Cheng |
| contents | We introduce an abstract neural flow framework for neural networks and neural operators. The framework contains two continuous-depth models, namely neural flows with composition and separation structures, and covers both finite-dimensional function approximation and infinite-dimensional operator approximation. We prove well-posedness and universal approximation properties for the corresponding neural flows, including, to the best of our knowledge, the first universal approximation result for flow-based models between infinite-dimensional spaces. We also obtain universal approximation results for convolutional neural flow models. Through suitable time discretizations, the composition structure recovers ResNet-type architectures, while the separation structure, via a splitting-based discretization, yields plain architectures. This gives a unified flow-based route to both residual and plain architectures for neural networks and neural operators with fully connected or convolutional linear layers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_22557 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations Chen, Shuang He, Juncai Tai, Xue-Cheng Machine Learning Numerical Analysis 65L05, 68T07, 47J35, 41A65, 41A46 We introduce an abstract neural flow framework for neural networks and neural operators. The framework contains two continuous-depth models, namely neural flows with composition and separation structures, and covers both finite-dimensional function approximation and infinite-dimensional operator approximation. We prove well-posedness and universal approximation properties for the corresponding neural flows, including, to the best of our knowledge, the first universal approximation result for flow-based models between infinite-dimensional spaces. We also obtain universal approximation results for convolutional neural flow models. Through suitable time discretizations, the composition structure recovers ResNet-type architectures, while the separation structure, via a splitting-based discretization, yields plain architectures. This gives a unified flow-based route to both residual and plain architectures for neural networks and neural operators with fully connected or convolutional linear layers. |
| title | Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations |
| topic | Machine Learning Numerical Analysis 65L05, 68T07, 47J35, 41A65, 41A46 |
| url | https://arxiv.org/abs/2605.22557 |