mlsanityData to 7 Oct 2026

Neural Ordinary Differential Equations on Manifolds

arXiv 2006.06663 ↗First posted 11 Jun 2020stat.ML cs.LGcited by 31

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Abstract (arXiv)

Normalizing flows are a powerful technique for obtaining reparameterizable samples from complex multimodal distributions. Unfortunately current approaches fall short when the underlying space has a non trivial topology, and are only available for the most basic geometries. Recently normalizing flows in Euclidean space based on Neural ODEs show great promise, yet suffer the same limitations. Using ideas from differential geometry and geometric control theory, we describe how neural ODEs can be extended to smooth manifolds. We show how vector fields provide a general framework for parameterizing a flexible class of invertible mapping on these spaces and we illustrate how gradient based learning can be performed. As a result we define a general methodology for building normalizing flows on manifolds.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). First posted on arXiv 11 Jun 2020.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 6; 2022: 3; 2023: 7; 2024: 5; 2025: 2; 2026: 8.

31 papers.

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  1. 2022Riemannian Score-Based Generative Modellingcited by 216
  2. 2022Riemannian Diffusion Modelscited by 120
  3. 2024FlowMM: Generating Materials with Riemannian Flow Matchingcited by 105
  4. 2021Moser Flow: Divergence-based Generative Modeling on Manifoldscited by 67
  5. 2022Matching Normalizing Flows and Probability Paths on Manifoldscited by 57
  6. 2021Introduction to Normalizing Flows for Lattice Field Theorycited by 35
  7. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processescited by 21
  8. 2023Riemannian Residual Neural Networkscited by 21
  9. 2021Equivariant Manifold Flowscited by 20
  10. 2024Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groupscited by 19
  11. 2023Scaling Riemannian Diffusion Modelscited by 18
  12. 2023Delving into Discrete Normalizing Flows on SO(3) Manifold for Probabilistic Rotation Modelingcited by 14
19 more
  1. 2023A Review of Change of Variable Formulas for Generative Modelingcited by 12
  2. 2021Density estimation on smooth manifolds with normalizing flowscited by 10
  3. 2021Deep Efficient Continuous Manifold Learning for Time Series Modelingcited by 10
  4. 2025Generalised Flow Maps for Few-Step Generative Modelling on Riemannian Manifoldscited by 10
  5. 2023Learning Distributions on Manifolds with Free-Form Flowscited by 7
  6. 2023Learning on Manifolds: Universal Approximations Properties using Geometric Controllability Conditions for Neural ODEscited by 4
  7. 2024Non-Perturbative Trivializing Flows for Lattice Gauge Theoriescited by 3
  8. 2021Learning ODEs via Diffeomorphisms for Fast and Robust Integrationcited by 2
  9. 2024Equivariant Manifold Neural ODEs and Differential Invariantscited by 2
  10. 2025Are Euler angles a useful rotation parameterisation for pose estimation with Normalizing Flows?cited by 2
  11. 2026Manifold Aware Denoising Score Matching (MAD)cited by 1
  12. 2024Optimal Potential Shaping on SE(3) via Neural ODEs on Lie Groupscited by 0
  13. 2026Geometry-Preserving Neural Architectures on Manifolds with Boundarycited by 0
  14. 2026PolyNODE: Variable-dimension Neural ODEs on M-polyfoldscited by 0
  15. 2026An Invariant Compiler for Neural ODEs in AI-Accelerated Scientific Simulationcited by 0
  16. 2026Steerable Neural ODEs on Homogeneous Spacescited by 0
  17. 2026Learning Unbiased Permutations via Flow Matchingcited by 0
  18. 2026Riemannian Neural Hamiltonian Flows: Geodesic Symplectic Transport and Interpretabilitycited by 0
  19. 2026Riemannian Flow Models with Reinforcement Learning for Molecular Crystal Structure Predictioncited by 0