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Neural Manifold Ordinary Differential Equations

arXiv 2006.10254 ↗First posted 18 Jun 2020stat.ML cs.LG math.DGcited by 81

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Abstract (arXiv)

To better conform to data geometry, recent deep generative modelling techniques adapt Euclidean constructions to non-Euclidean spaces. In this paper, we study normalizing flows on manifolds. Previous work has developed flow models for specific cases; however, these advancements hand craft layers on a manifold-by-manifold basis, restricting generality and inducing cumbersome design constraints. We overcome these issues by introducing Neural Manifold Ordinary Differential Equations, a manifold generalization of Neural ODEs, which enables the construction of Manifold Continuous Normalizing Flows (MCNFs). MCNFs require only local geometry (therefore generalizing to arbitrary manifolds) and compute probabilities with continuous change of variables (allowing for a simple and expressive flow construction). We find that leveraging continuous manifold dynamics produces a marked improvement for both density estimation and downstream tasks.

Key facts. Cited by 81 papers in the map (data through 7 Oct 2026). First posted on arXiv 18 Jun 2020.

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Citing papers per year: 2021: 14; 2022: 10; 2023: 15; 2024: 12; 2025: 15; 2026: 15.

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  1. 2022Diffusion Models: A Comprehensive Survey of Methods and Applicationscited by 1,107
  2. 2023Flow Matching on General Geometriescited by 260
  3. 2022Riemannian Score-Based Generative Modellingcited by 216
  4. 2022Riemannian Diffusion Modelscited by 120
  5. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 74
  6. 2021Moser Flow: Divergence-based Generative Modeling on Manifoldscited by 67
  7. 2022Matching Normalizing Flows and Probability Paths on Manifoldscited by 57
  8. 2021Smooth Normalizing Flowscited by 52
  9. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraintscited by 46
  10. 2023Rigid Body Flows for Sampling Molecular Crystal Structurescited by 27
  11. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processescited by 21
  12. 2023Riemannian Residual Neural Networkscited by 21
38 more
  1. 2021Equivariant Manifold Flowscited by 20
  2. 2024Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groupscited by 19
  3. 2023Scaling Riemannian Diffusion Modelscited by 18
  4. 2022Learning Riemannian Stable Dynamical Systems via Diffeomorphismscited by 17
  5. 2021Riemannian Convex Potential Mapscited by 16
  6. 2022Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projectionscited by 16
  7. 2023Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraintscited by 15
  8. 2024Spiking Graph Neural Network on Riemannian Manifoldscited by 15
  9. 2026Language Modeling with Hyperspherical Flowscited by 15
  10. 2023Delving into Discrete Normalizing Flows on SO(3) Manifold for Probabilistic Rotation Modelingcited by 14
  11. 2025Flow Matching Meets Biology and Life Science: A Surveycited by 14
  12. 2021Learning Stable Vector Fields on Lie Groupscited by 12
  13. 2021Density estimation on low-dimensional manifolds: an inflation-deflation approachcited by 11
  14. 2021Second-Order Neural ODE Optimizercited by 11
  15. 2021Continuous normalizing flows on manifoldscited by 10
  16. 2021Density estimation on smooth manifolds with normalizing flowscited by 10
  17. 2021Deep Efficient Continuous Manifold Learning for Time Series Modelingcited by 10
  18. 2025Generalised Flow Maps for Few-Step Generative Modelling on Riemannian Manifoldscited by 10
  19. 2024Generative Modeling of Discrete Joint Distributions by E-Geodesic Flow Matching on Assignment Manifoldscited by 9
  20. 2021Continuous-Depth Neural Models for Dynamic Graph Predictioncited by 7
  21. 2023Learning Distributions on Manifolds with Free-Form Flowscited by 7
  22. 2023Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learningcited by 6
  23. 2022Nonlinear Isometric Manifold Learning for Injective Normalizing Flowscited by 5
  24. 2024Projected Neural Differential Equations for Learning Constrained Dynamicscited by 5
  25. 2026Riemannian MeanFlowcited by 5
  26. 2021Manifold Density Estimation via Generalized Dequantizationcited by 4
  27. 2022Learning Subgrid-scale Models with Neural Ordinary Differential Equationscited by 4
  28. 2023Learning on Manifolds: Universal Approximations Properties using Geometric Controllability Conditions for Neural ODEscited by 4
  29. 2023Deep Geometric Learning with Monotonicity Constraints for Alzheimer's Disease Progressioncited by 4
  30. 2025Riemannian Denoising Diffusion Probabilistic Modelscited by 4
  31. 2022Grassmann Manifold Flows for Stable Shape Generationcited by 3
  32. 2023Learning to Decouple Complex Systemscited by 3
  33. 2024Non-Perturbative Trivializing Flows for Lattice Gauge Theoriescited by 3
  34. 2021Learning ODEs via Diffeomorphisms for Fast and Robust Integrationcited by 2
  35. 2024Equivariant Manifold Neural ODEs and Differential Invariantscited by 2
  36. 2024R-ODE: Ricci Curvature Tells When You Will be Informedcited by 2
  37. 2025Generative Machine Learning for Multivariate Angular Simulationcited by 2
  38. 2025Efficient Manifold-Constrained Neural ODE for High-Dimensional Datasetscited by 2

50 most cited of 81 All 81 (JSON)