mlsanityData to 7 Oct 2026

Rectified Flow: A Marginal Preserving Approach to Optimal Transport

arXiv 2209.14577 ↗First posted 29 Sep 2022stat.ML cs.LGcited by 277

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

Abstract (arXiv)

We present a flow-based approach to the optimal transport (OT) problem between two continuous distributions π0,π1 on ℝd, of minimizing a transport cost mathbbE[c(X1-X0)] in the set of couplings (X0,X1) whose marginal distributions on X0,X1 equals π0,π1, respectively, where c is a cost function. Our method iteratively constructs a sequence of neural ordinary differentiable equations (ODE), each learned by solving a simple unconstrained regression problem, which monotonically reduce the transport cost while automatically preserving the marginal constraints. This yields a monotonic interior approach that traverses inside the set of valid couplings to decrease the transport cost, which distinguishes itself from most existing approaches that enforce the coupling constraints from the outside. The main idea of the method draws from rectified flow, a recent approach that simultaneously decreases the whole family of transport costs induced by convex functions c (and is hence multi-objective in nature), but is not tailored to minimize a specific transport cost. Our method is a single-object variant of rectified flow that guarantees to solve the OT problem for a fixed, user-specified convex cost function c.

Key facts. Cited by 277 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 9 of its 30 references that are in the map. First posted on arXiv 29 Sep 2022.

Builds on

9 of 30 references.

Closest ancestors · 1

Same direction, newest first

  1. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow7 Sep

Other references · 8

  1. 2020Denoising Diffusion Implicit Modelscited by 9,655
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  4. 2019Optimal transport mapping via input convex neural networkscited by 202
  5. 2022Neural Optimal Transportcited by 123
  6. 2021Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmarkcited by 95
  7. 2020Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimizationcited by 89
  8. 2016An introduction to sampling via measure transportcited by 16

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 4; 2023: 21; 2024: 45; 2025: 120; 2026: 87.

277 papers.

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  1. 2024π0: A Vision-Language-Action Flow Model for General Robot Controlcited by 2,297
  2. 2025π0.5: a Vision-Language-Action Model with Open-World Generalizationcited by 1,653
  3. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  4. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  5. 2023Improving and generalizing flow-based generative models with minibatch optimal transportcited by 835
  6. 2025SmolVLA: A Vision-Language-Action Model for Affordable and Efficient Roboticscited by 507
  7. 2023InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generationcited by 387
  8. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplingscited by 267
  9. 2024Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Controlcited by 222
  10. 2024Taming Rectified Flow for Inversion and Editingcited by 200
  11. 2023UFOGen: You Forward Once Large Scale Text-to-Image Generation via Diffusion GANscited by 165
  12. 2024JanusFlow: Harmonizing Autoregression and Rectified Flow for Unified Multimodal Understanding and Generationcited by 162
38 more
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  4. 2025GR-3 Technical Reportcited by 110
  5. 2024Improving the Training of Rectified Flowscited by 106
  6. 2025OneTwoVLA: A Unified Vision-Language-Action Model with Adaptive Reasoningcited by 105
  7. 2024PeRFlow: Piecewise Rectified Flow as Universal Plug-and-Play Acceleratorcited by 95
  8. 2022Action Matching: Learning Stochastic Dynamics from Samplescited by 94
  9. 2023Stochastic interpolants with data-dependent couplingscited by 94
  10. 2024PnP-Flow: Plug-and-Play Image Restoration with Flow Matchingcited by 92
  11. 2023Minimizing Trajectory Curvature of ODE-based Generative Modelscited by 87
  12. 2025ForceVLA: Enhancing VLA Models with a Force-aware MoE for Contact-rich Manipulationcited by 87
  13. 2025DriveMoE: Mixture-of-Experts for Vision-Language-Action Model in End-to-End Autonomous Drivingcited by 85
  14. 2023Simulation-free Schr"odinger bridges via score and flow matchingcited by 81
  15. 2023Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modelingcited by 80
  16. 2024Optimal Flow Matching: Learning Straight Trajectories in Just One Stepcited by 74
  17. 2025Energy-Weighted Flow Matching for Offline Reinforcement Learningcited by 70
  18. 2023On the Generalization Properties of Diffusion Modelscited by 67
  19. 2023Equivariant Flow Matching with Hybrid Probability Transportcited by 59
  20. 2025villa-X: Enhancing Latent Action Modeling in Vision-Language-Action Modelscited by 58
  21. 2022Fast Point Cloud Generation with Straight Flowscited by 56
  22. 2025Evo-0: Vision-Language-Action Model with Implicit Spatial Understandingcited by 55
  23. 2024Affordance-based Robot Manipulation with Flow Matchingcited by 54
  24. 2024AdaFlow: Imitation Learning with Variance-Adaptive Flow-Based Policiescited by 50
  25. 2024Schr"odinger Bridge Flow for Unpaired Data Translationcited by 47
  26. 2025KV-Edit: Training-Free Image Editing for Precise Background Preservationcited by 47
  27. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 43
  28. 2024Neural Flow Diffusion Models: Learnable Forward Process for Improved Diffusion Modellingcited by 43
  29. 2025GigaBrain-0: A World Model-Powered Vision-Language-Action Modelcited by 43
  30. 2026World Model for Robot Learning: A Comprehensive Surveycited by 42
  31. 2025H-RDT: Human Manipulation Enhanced Bimanual Robotic Manipulationcited by 41
  32. 2025GR-RL: Going Dexterous and Precise for Long-Horizon Robotic Manipulationcited by 41
  33. 2023Generalized Schr"odinger Bridge Matchingcited by 40
  34. 2022On the Power of Foundation Modelscited by 37
  35. 2024FlowIE: Efficient Image Enhancement via Rectified Flowcited by 37
  36. 2025CoMo: Learning Continuous Latent Motion from Internet Videos for Scalable Robot Learningcited by 37
  37. 2024The Superposition of Diffusion Models Using the Ito Density Estimatorcited by 36
  38. 2025Scalable Equilibrium Sampling with Sequential Boltzmann Generatorscited by 36

50 most cited of 277 All 277 (JSON)

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  1. 2025The Curse of Conditions: Analyzing and Improving Optimal Transport for Conditional Flow-Based Generation
  2. 2026Order-Optimal Sample Complexity of Rectified Flows
  3. 2026Beyond Transport Cost: Routing Differences between Flow Matching and Optimal Transport
  4. 2026Expected Batch Optimal Transport Plans and Consequences for Flow Matching
  5. 2026Optimal Transport Flow Matching by Design
  6. 2026Branched Optimal Transport Amortization
  7. 2026The Fundamental Limits of Valid Transport Map Estimation
  8. 2021Score-based Generative Neural Networks for Large-Scale Optimal Transport