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arXiv · 2605.22557

Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations

Shuang Chen·Juncai He·Xue-Cheng Tai
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Abstract

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.

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Shuang Chen, Juncai He, Xue-Cheng Tai. 2026-05-26. Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations. https://arxiv.org/abs/2605.22557

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