mlsanityData to 14 Oct 2026

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

arXiv 2605.22557 ↗First posted 21 May 2026cs.LG cs.NA math.NAcited by 0

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Abstract (arXiv)

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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References · 55

  1. 2015Deep Residual Learning for Image Recognitioncited by 52,096
  2. 2022Flow Matching for Generative Modelingcited by 5,702
  3. 2018Neural Ordinary Differential Equationscited by 3,779
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  5. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  6. 2016Error bounds for approximations with deep ReLU networkscited by 574
  7. 2025Mean Flows for One-step Generative Modelingcited by 561
  8. 2014On the Number of Linear Regions of Deep Neural Networkscited by 381
  9. 2021Choose a Transformer: Fourier or Galerkincited by 343
  10. 2014Breaking the Curse of Dimensionality with Convex Neural Networkscited by 331
  11. 2024Transolver: A Fast Transformer Solver for PDEs on General Geometriescited by 326
  12. 2017The Expressive Power of Neural Networks: A View from the Widthcited by 320
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  1. 2017Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networkscited by 311
  2. 2016Benefits of depth in neural networkscited by 250
  3. 2017Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equationscited by 203
  4. 2019Deep Network Approximation Characterized by Number of Neuronscited by 122
  5. 2021Optimal Approximation Rate of ReLU Networks in terms of Width and Depthcited by 96
  6. 2021Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networkscited by 86
  7. 2016Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with ℓ1 and ℓ0 Controlscited by 84
  8. 2018ResNet with one-neuron hidden layers is a Universal Approximatorcited by 79
  9. 2019Deep Learning via Dynamical Systems: An Approximation Perspectivecited by 78
  10. 2018A Priori Estimates of the Population Risk for Two-layer Neural Networkscited by 76
  11. 2023Neural Inverse Operators for Solving PDE Inverse Problemscited by 65
  12. 2016Deep vs. shallow networks : An approximation theory perspectivecited by 63
  13. 2017Multi-level Residual Networks from Dynamical Systems Viewcited by 59
  14. 2022Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spacescited by 51
  15. 2022Mitigating spectral bias for the multiscale operator learningcited by 44
  16. 2019Approximation and Non-parametric Estimation of ResNet-type Convolutional Neural Networkscited by 40
  17. 2023MgNO: Efficient Parameterization of Linear Operators via Multigridcited by 40
  18. 2019The Barron Space and the Flow-induced Function Spaces for Neural Network Modelscited by 39
  19. 2018Equivalence of approximation by convolutional neural networks and fully-connected networkscited by 38
  20. 2021Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow modelscited by 35
  21. 2023Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivativescited by 30
  22. 2019Deep neural networks, generic universal interpolation, and controlled ODEscited by 28
  23. 2022Achieve the Minimum Width of Neural Networks for Universal Approximationcited by 26
  24. 2023PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networkscited by 17
  25. 2023Interpolation, Approximation and Controllability of Deep Neural Networkscited by 16
  26. 2021Approximation Properties of Deep ReLU CNNscited by 15
  27. 2023Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal Approximationcited by 13
  28. 2021ReLU Deep Neural Networks from the Hierarchical Basis Perspectivecited by 12
  29. 2023Adaptive operator learning for infinite-dimensional Bayesian inverse problemscited by 11
  30. 2023Expressivity and Approximation Properties of Deep Neural Networks with ReLUk Activationcited by 9
  31. 2024Interplay between depth and width for interpolation in neural ODEscited by 9
  32. 2023Deep Neural Networks and Finite Elements of Any Order on Arbitrary Dimensionscited by 8
  33. 2022Approximation analysis of CNNs from a feature extraction viewcited by 6
  34. 2025Deep Neural Networks with General Activations: Super-Convergence in Sobolev Normscited by 6
  35. 2024A Mathematical Explanation of UNetcited by 4
  36. 2023A Minimal Control Family of Dynamical Systems for Universal Approximationcited by 3
  37. 2025Self-composing neural operators for high-frequency and multiscale PDE surrogatescited by 3
  38. 2025Deep Neural ODE Operator Networks for PDEscited by 3

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