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

2026-05-21

AI Summary

This work proposes a unified theoretical framework for neural networks capable of approximating both finite-dimensional functions and infinite-dimensional operators. To this end, an abstract neural flow model is introduced, encompassing both compositional and decoupled continuous-depth architectures, which naturally connects ResNet-like and plain networks through temporal discretization. For the first time, the universal approximation capability of such flow-based models is rigorously established between infinite-dimensional Banach spaces. The study develops a comprehensive theory addressing well-posedness and approximation properties, applicable to both fully connected and convolutional architectures. This provides a unified continuous-depth perspective and a solid foundation in functional analysis for learning both functions and operators.

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.

Problem

• Neural Operators
• Universal Approximation
• Infinite-dimensional Spaces
• Neural Networks
• Flow-based Models

Innovation

• Neural Flow Operators
• Universal Approximation
• Infinite-dimensional Operators
• Continuous-depth Models
• ResNet Unification