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Steerable Neural ODEs on Homogeneous Spaces

arXiv 2605.11133 ↗First posted 11 May 2026cs.LG math.DGcited by 0

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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.

Key facts. No citing papers in the map yet (data through 13 Oct 2026). Builds on 24 of its 34 references that are in the map. First posted on arXiv 11 May 2026.

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References · 24

  1. 2022Flow Matching for Generative Modelingcited by 5,542
  2. 2018Neural Ordinary Differential Equationscited by 3,750
  3. 2016Steerable CNNscited by 300
  4. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 297
  5. 2020Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densitiescited by 250
  6. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 225
  7. 2021E(n) Equivariant Normalizing Flowscited by 165
  8. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generationcited by 148
  9. 2020Riemannian Continuous Normalizing Flowscited by 145
  10. 2020Neural Manifold Ordinary Differential Equationscited by 81
  11. 2020Sheaf Neural Networkscited by 81
  12. 2021Geometric Deep Learning and Equivariant Neural Networkscited by 64
12 more
  1. 2022Sheaf Neural Networks with Connection Laplacianscited by 64
  2. 2020Lattice gauge equivariant convolutional neural networkscited by 45
  3. 2020Neural Ordinary Differential Equations on Manifoldscited by 31
  4. 2025La-Proteina: Atomistic Protein Generation via Partially Latent Flow Matchingcited by 28
  5. 2021Homogeneous vector bundles and G-equivariant convolutional neural networkscited by 21
  6. 2021Equivariant Manifold Flowscited by 20
  7. 2025FlowMol3: Flow Matching for 3D De Novo Small-Molecule Generationcited by 20
  8. 2019Covariance in Physics and Convolutional Neural Networkscited by 12
  9. 2025PEAR: Equal Area Weather Forecasting on the Spherecited by 5
  10. 2025Equivariant non-linear maps for neural networks on homogeneous spacescited by 4
  11. 2024Equivariant Manifold Neural ODEs and Differential Invariantscited by 2
  12. 2025Learning Chern Numbers of Topological Insulators with Gauge Equivariant Neural Networkscited by 2

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