mlsanityData to 7 Oct 2026

Autonomous-Flow-Based Generation

arXiv 2511.09902 ↗First posted 13 Nov 2025cs.LG cs.NA math.CAcited by 1

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Abstract (arXiv)

We show that using autonomous-flow-based generation, one can universally approximate orientation-preserving diffeomorphisms defined on the cube by Neural ODEs with rate O(P-1/d) with P parameters. On the other hand, we show that by using only a single autonomous flow, the class of Neural ODEs is nowhere dense on the cube in dimension d ≥ 2 . Under a compact-supportid condition on (0,1)d, we show that using autonomous-flow-based generation, one can universally approximate compactly supportedid diffeomorphisms on (0,1)d for any dimension with rate O((fracPlog P)-2/d) with P parameters and for compactly supportedid homeomorphisms on (0,1)d in dimension d ≥ 5 with rate O(P-1/(d+1)) with P parameters and by a composition of at most Id autonomous Neural ODEs with the same supportid, where Id depends only on the dimension. Moreover, we show that the class of single autonomous flows compactly supportedid on (0,1)d is meagre in the space of compactly supportedid homeomorphisms on (0,1)d for d≥ 2. By linearly lifting the domain into one higher dimension, we obtain a universal approximation result for Lipschitz functions compactly supported on (0,1)d with rate O(P-1/(d+1)) with P parameters.

Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 31 of its 86 references that are in the map. First posted on arXiv 13 Nov 2025.

Builds on

31 of 86 references.

References · 31

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  2. 2016Group Equivariant Convolutional Networkscited by 1,154
  3. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
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19 more
  1. 2019Approximation Capabilities of Neural ODEs and Invertible Residual Networkscited by 56
  2. 2022Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spacescited by 50
  3. 2021Universal Approximation Theorems for Differentiable Geometric Deep Learningcited by 41
  4. 2021Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neuronscited by 41
  5. 2024Log Neural Controlled Differential Equations: The Lie Brackets Make a Differencecited by 34
  6. 2022Universal approximation property of invertible neural networkscited by 30
  7. 2020Non-Euclidean Universal Approximationcited by 29
  8. 2019Deep neural networks, generic universal interpolation, and controlled ODEscited by 28
  9. 2023Neural signature kernels as infinite-width-depth-limits of controlled ResNetscited by 25
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  11. 2023Interpolation, Approximation and Controllability of Deep Neural Networkscited by 15
  12. 2024Lecture notes on rough paths and applications to machine learningcited by 13
  13. 2021Deep Learning Approximation of Diffeomorphisms via Linear-Control Systemscited by 7
  14. 2024Bridging the Gap Between Approximation and Learning via Optimal Approximation by ReLU MLPs of Maximal Regularitycited by 7
  15. 2022Do ReLU Networks Have An Edge When Approximating Compactly-Supported Functions?cited by 6
  16. 2022Fitting an immersed submanifold to data via Sussmann's orbit theoremcited by 4
  17. 2025Approximation properties of neural ODEscited by 2
  18. 2020A method for large diffeomorphic registration via broken geodesicscited by 1
  19. 2024Stationary Velocity Fields on Matrix Groups for Deformable Image Registrationcited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processescited by 0