# Learning on Manifolds: Universal Approximations Properties using Geometric Controllability Conditions for Neural ODEs

arXiv: https://arxiv.org/abs/2305.08849 · HTML: https://mlsanity.com/paper/2305.08849/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 16 references that are in the map.
- First posted on arXiv 15 May 2023.

## Builds on (most cited first, 9 of 9)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Neural Ordinary Differential Equations on Manifolds](https://mlsanity.com/paper/2006.06663/) (2020)
- [Universal Approximation Power of Deep Residual Neural Networks via Nonlinear Control Theory](https://mlsanity.com/paper/2007.06007/) (2020)
- [Neural ODE Control for Trajectory Approximation of Continuity Equation](https://mlsanity.com/paper/2205.09241/) (2022)
- [Optimal Control of Continuity Equations](https://mlsanity.com/paper/1506.08932/) (2015)

## Built on by (most cited first, 5 of 5)

- [Cluster-based classification with neural ODEs via control](https://mlsanity.com/paper/2312.13807/) (2023)
- [Universal approximation property of ODENet and ResNet with a single activation function](https://mlsanity.com/paper/2410.16709/) (2024)
- [Exact Fixed-Point Constraints in Neural-ODEs with Provable Universality](https://mlsanity.com/paper/2605.10613/) (2026)
- [Geometry-Preserving Neural Architectures on Manifolds with Boundary](https://mlsanity.com/paper/2602.03082/) (2026)
- [Deep learning from a control vantage point](https://mlsanity.com/paper/2610.14385/) (2026)

## Abstract (arXiv)

In numerous robotics and mechanical engineering applications, among others, data is often constrained on smooth manifolds due to the presence of rotational degrees of freedom. Common datadriven and learning-based methods such as neural ordinary differential equations (ODEs), however, typically fail to satisfy these manifold constraints and perform poorly for these applications. To address this shortcoming, in this paper we study a class of neural ordinary differential equations that, by design, leave a given manifold invariant, and characterize their properties by leveraging the controllability properties of control affine systems. In particular, using a result due to Agrachev and Caponigro on approximating diffeomorphisms with flows of feedback control systems, we show that any map that can be represented as the flow of a manifold-constrained dynamical system can also be approximated using the flow of manifold-constrained neural ODE, whenever a certain controllability condition is satisfied. Additionally, we show that this universal approximation property holds when the neural ODE has limited width in each layer, thus leveraging the depth of network instead for approximation. We verify our theoretical findings using numerical experiments on PyTorch for the manifolds S2 and the 3-dimensional orthogonal group SO(3), which are model manifolds for mechanical systems such as spacecrafts and satellites. We also compare the performance of the manifold invariant neural ODE with classical neural ODEs that ignore the manifold invariant properties and show the superiority of our approach in terms of accuracy and sample complexity.