# Autonomous-Flow-Based Generation

arXiv: https://arxiv.org/abs/2511.09902 · HTML: https://mlsanity.com/paper/2511.09902/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 86 references that are in the map.
- First posted on arXiv 13 Nov 2025.

## Builds on (most cited first, 30 of 31)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Deep vs. shallow networks : An approximation theory perspective](https://mlsanity.com/paper/1608.03287/) (2016)
- [Approximation Capabilities of Neural ODEs and Invertible Residual Networks](https://mlsanity.com/paper/1907.12998/) (2019)
- [Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces](https://mlsanity.com/paper/2211.14400/) (2022)
- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons](https://mlsanity.com/paper/2107.02397/) (2021)
- [Log Neural Controlled Differential Equations: The Lie Brackets Make a Difference](https://mlsanity.com/paper/2402.18512/) (2024)
- [Universal approximation property of invertible neural networks](https://mlsanity.com/paper/2204.07415/) (2022)
- [Non-Euclidean Universal Approximation](https://mlsanity.com/paper/2006.02341/) (2020)
- [Deep neural networks, generic universal interpolation, and controlled ODEs](https://mlsanity.com/paper/1908.07838/) (2019)
- [Neural signature kernels as infinite-width-depth-limits of controlled ResNets](https://mlsanity.com/paper/2303.17671/) (2023)
- [Mathematical theory of deep learning](https://mlsanity.com/paper/2407.18384/) (2024)
- [Interpolation, Approximation and Controllability of Deep Neural Networks](https://mlsanity.com/paper/2309.06015/) (2023)
- [Lecture notes on rough paths and applications to machine learning](https://mlsanity.com/paper/2404.06583/) (2024)
- [Bridging the Gap Between Approximation and Learning via Optimal Approximation by ReLU MLPs of Maximal Regularity](https://mlsanity.com/paper/2409.12335/) (2024)
- [Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems](https://mlsanity.com/paper/2110.12393/) (2021)
- [Do ReLU Networks Have An Edge When Approximating Compactly-Supported Functions?](https://mlsanity.com/paper/2204.11231/) (2022)
- [Fitting an immersed submanifold to data via Sussmann's orbit theorem](https://mlsanity.com/paper/2204.01119/) (2022)
- [Approximation properties of neural ODEs](https://mlsanity.com/paper/2503.15696/) (2025)
- [A method for large diffeomorphic registration via broken geodesics](https://mlsanity.com/paper/2011.14298/) (2020)

## Built on by (most cited first, 1 of 1)

- [NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes](https://mlsanity.com/paper/2607.14361/) (2026)

## 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.