mlsanityData to 7 Oct 2026

Flow Matching on General Geometries

arXiv 2302.03660 ↗First posted 7 Feb 2023cs.LG cs.AI stat.MLcited by 260

On the map: Diffusion models → Flow matching, time series and science

Abstract (arXiv)

We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, are inherently unable to scale to high dimensions, or use approximations for limiting quantities that result in biased training objectives. Riemannian Flow Matching bypasses these limitations and offers several advantages over previous approaches: it is simulation-free on simple geometries, does not require divergence computation, and computes its target vector field in closed-form. The key ingredient behind RFM is the construction of a relatively simple premetric for defining target vector fields, which encompasses the existing Euclidean case. To extend to general geometries, we rely on the use of spectral decompositions to efficiently compute premetrics on the fly. Our method achieves state-of-the-art performance on many real-world non-Euclidean datasets, and we demonstrate tractable training on general geometries, including triangular meshes with highly non-trivial curvature and boundaries.

Key facts. Cited by 260 papers in the map (data through 7 Oct 2026). Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. Builds on 22 of its 62 references that are in the map. First posted on arXiv 7 Feb 2023.

Builds on

22 of 62 references.

Closest ancestors · 2

Same direction, newest first

  1. 2022Flow Matching for Generative Modeling6 Oct
  2. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow7 Sep

Other references · 20

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  4. 2018Neural Ordinary Differential Equationscited by 3,733
  5. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  6. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 584
  7. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 381
  8. 2020Scalable Gradients for Stochastic Differential Equationscited by 264
  9. 2022Riemannian Score-Based Generative Modellingcited by 216
  10. 2020Riemannian Continuous Normalizing Flowscited by 143
  11. 2022Riemannian Diffusion Modelscited by 120
  12. 2020Normalizing Flows on Tori and Spherescited by 117
8 more
  1. 2020Neural Manifold Ordinary Differential Equationscited by 81
  2. 2016Normalizing Flows on Riemannian Manifoldscited by 68
  3. 2021Moser Flow: Divergence-based Generative Modeling on Manifoldscited by 67
  4. 2022Matching Normalizing Flows and Probability Paths on Manifoldscited by 57
  5. 2020Latent Variable Modelling with Hyperbolic Normalizing Flowscited by 49
  6. 2018Spectral Inference Networks: Unifying Deep and Spectral Learningcited by 34
  7. 2022Neural Eigenfunctions Are Structured Representation Learnerscited by 16
  8. 2021Continuous normalizing flows on manifoldscited by 10

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 12; 2024: 31; 2025: 96; 2026: 121.

260 papers.

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  1. 2024Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Designcited by 294
  2. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplingscited by 267
  3. 2023Flow Matching in Latent Spacecited by 155
  4. 2023Diffusion Schr"odinger Bridge Matchingcited by 151
  5. 2024Dirichlet Flow Matching with Applications to DNA Sequence Designcited by 110
  6. 2024Phased Consistency Modelscited by 83
  7. 2025Inference-Time Alignment in Diffusion Models with Reward-Guided Generation: Tutorial and Reviewcited by 82
  8. 2024Variational Flow Matching for Graph Generationcited by 78
  9. 2024Learning Robotic Manipulation Policies from Point Clouds with Conditional Flow Matchingcited by 63
  10. 2025Proteina: Scaling Flow-based Protein Structure Generative Modelscited by 63
  11. 2024Riemannian Flow Matching Policy for Robot Motion Learningcited by 61
  12. 2024Mixed Continuous and Categorical Flow Matching for 3D De Novo Molecule Generationcited by 53
38 more
  1. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  2. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraintscited by 46
  3. 2025On the Guidance of Flow Matchingcited by 44
  4. 2023Motion Flow Matching for Human Motion Synthesis and Editingcited by 36
  5. 2024Sequence-Augmented SE(3)-Flow Matching For Conditional Protein Backbone Generationcited by 36
  6. 2025An Introduction to Flow Matching and Diffusion Modelscited by 33
  7. 2025Continuous Diffusion Model for Language Modelingcited by 29
  8. 2025Open Materials Generation with Stochastic Interpolantscited by 26
  9. 2024ActionFlow: Equivariant, Accurate, and Efficient Policies with Spatially Symmetric Flow Matchingcited by 25
  10. 2025The Evolution of Dataset Distillation: Toward Scalable and Generalizable Solutionscited by 24
  11. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processescited by 21
  12. 2024Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groupscited by 19
  13. 2025Diffuse Everything: Multimodal Diffusion Models on Arbitrary State Spacescited by 19
  14. 2023Gaussian Interpolation Flowscited by 17
  15. 2025Hyperspherical Latents Improve Continuous-Token Autoregressive Generationcited by 16
  16. 2024MOFFlow: Flow Matching for Structure Prediction of Metal-Organic Frameworkscited by 15
  17. 2025FlowMP: Learning Motion Fields for Robot Planning with Conditional Flow Matchingcited by 15
  18. 2025Exponential Family Variational Flow Matching for Tabular Data Generationcited by 15
  19. 2025Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularitycited by 14
  20. 2025Flow Matching Meets Biology and Life Science: A Surveycited by 14
  21. 2023Star-Shaped Denoising Diffusion Probabilistic Modelscited by 13
  22. 2024Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifoldscited by 13
  23. 2025FlowRAM: Grounding Flow Matching Policy with Region-Aware Mamba Framework for Robotic Manipulationcited by 13
  24. 2025Carr'e du champ flow matching: better quality-generalisation tradeoff in generative modelscited by 13
  25. 2024Multi-Modal and Multi-Attribute Generation of Single Cells with CFGencited by 12
  26. 2026Learning on the Manifold: Unlocking Standard Diffusion Transformers with Representation Encoderscited by 12
  27. 2023Manifold Diffusion Fieldscited by 11
  28. 2025Source-Guided Flow Matchingcited by 11
  29. 2025Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometrycited by 10
  30. 2025All that structure matches does not glittercited by 10
  31. 2026Phantom: Physics-Infused Video Generation via Joint Modeling of Visual and Latent Physical Dynamicscited by 10
  32. 2024Generative Modeling of Discrete Joint Distributions by E-Geodesic Flow Matching on Assignment Manifoldscited by 9
  33. 2024Markovian Flow Matching: Accelerating MCMC with Continuous Normalizing Flowscited by 9
  34. 2024Stable Consistency Tuning: Understanding and Improving Consistency Modelscited by 9
  35. 2025Riemannian Variational Flow Matching for Material and Protein Designcited by 9
  36. 2025Theoretical Guarantees for High Order Trajectory Refinement in Generative Flowscited by 9
  37. 2025An All-Atom Generative Model for Designing Protein Complexescited by 9
  38. 2025On the minimax optimality of Flow Matching through the connection to kernel density estimationcited by 9

50 most cited of 260 All 260 (JSON)

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Close in wording, no citation either way

  1. 2026Total Variation Rates for Riemannian Flow Matching
  2. 2025Generalised Flow Maps for Few-Step Generative Modelling on Riemannian Manifolds
  3. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifold
  4. 2024Categorical Flow Matching on Statistical Manifolds
  5. 2026When Riemann flows with Wasserstein: Generative Modeling of Probability Distributions on Manifolds
  6. 2025On the continuity of flows
  7. 2026MeshFlow: Mesh Generation with Equivariant Flow Matching
  8. 2026Robot Learning on Discrete Surfaces: Theory and Applications ↗