# Steerable Neural ODEs on Homogeneous Spaces

arXiv: https://arxiv.org/abs/2605.11133 · HTML: https://mlsanity.com/paper/2605.11133/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 34 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 24 of 24)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [Lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2012.12901/) (2020)
- [Neural Ordinary Differential Equations on Manifolds](https://mlsanity.com/paper/2006.06663/) (2020)
- [La-Proteina: Atomistic Protein Generation via Partially Latent Flow Matching](https://mlsanity.com/paper/2507.09466/) (2025)
- [Homogeneous vector bundles and G-equivariant convolutional neural networks](https://mlsanity.com/paper/2105.05400/) (2021)
- [FlowMol3: Flow Matching for 3D De Novo Small-Molecule Generation](https://mlsanity.com/paper/2508.12629/) (2025)
- [Equivariant Manifold Flows](https://mlsanity.com/paper/2107.08596/) (2021)
- [Covariance in Physics and Convolutional Neural Networks](https://mlsanity.com/paper/1906.02481/) (2019)
- [PEAR: Equal Area Weather Forecasting on the Sphere](https://mlsanity.com/paper/2505.17720/) (2025)
- [Equivariant non-linear maps for neural networks on homogeneous spaces](https://mlsanity.com/paper/2504.20974/) (2025)
- [Equivariant Manifold Neural ODEs and Differential Invariants](https://mlsanity.com/paper/2401.14131/) (2024)
- [Learning Chern Numbers of Topological Insulators with Gauge Equivariant Neural Networks](https://mlsanity.com/paper/2502.15376/) (2025)

## Abstract (arXiv)

We introduce steerable neural ordinary differential equations on homogeneous spaces M=G/H. These models constitute a novel geometric extension of manifold neural ordinary differential equations (NODEs) that transport associated feature vectors transforming under the local symmetry group H. We interpret features as sections of associated vector bundles over M, and describe their evolution as parallel transport. This results in a coupled system of ODEs consisting of a flow equation on M and a steering equation acting on features. We show that steerable NODEs are G-equivariant whenever the vector field generating the flow and the connection governing parallel transport are both G-invariant. Furthermore, we demonstrate how steerable NODEs incorporate existing NODE models and continuous normalizing flows on Lie groups. Our framework provides the geometric foundation for learning continuous-time equivariant dynamics of general vector-valued features on homogeneous spaces.