mlsanityData to 7 Oct 2026

Optimal Flow Matching: Learning Straight Trajectories in Just One Step

arXiv 2403.13117 ↗First posted 19 Mar 2024stat.ML cs.LGcited by 74

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

Abstract (arXiv)

Over the several recent years, there has been a boom in development of Flow Matching (FM) methods for generative modeling. One intriguing property pursued by the community is the ability to learn flows with straight trajectories which realize the Optimal Transport (OT) displacements. Straightness is crucial for the fast integration (inference) of the learned flow's paths. Unfortunately, most existing flow straightening methods are based on non-trivial iterative FM procedures which accumulate the error during training or exploit heuristics based on minibatch OT. To address these issues, we develop and theoretically justify the novel Optimal Flow Matching (OFM) approach which allows recovering the straight OT displacement for the quadratic transport in just one FM step. The main idea of our approach is the employment of vector field for FM which are parameterized by convex functions.

Key facts. Cited by 74 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 54 of its 74 references that are in the map. First posted on arXiv 19 Mar 2024.

Builds on

54 of 74 references.

Closest ancestors · 6

Same direction, newest first

  1. 2023InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation12 Sep
  2. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplings28 Apr
  3. 2023Improving and generalizing flow-based generative models with minibatch optimal transport1 Feb
  4. 2022Flow Matching for Generative Modeling6 Oct
  5. 2022Rectified Flow: A Marginal Preserving Approach to Optimal Transport29 Sep
  6. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow7 Sep

Other references · 48

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  4. 2021Convex optimizationcited by 2,479
  5. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,277
  6. 2016An overview of gradient descent optimization algorithmscited by 1,021
  7. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modelingcited by 594
  8. 2016Input Convex Neural Networkscited by 407
  9. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 286
  10. 2022A Survey on Generative Diffusion Modelcited by 238
  11. 2019Optimal transport mapping via input convex neural networkscited by 202
  12. 2016Stochastic Optimization for Large-scale Optimal Transportcited by 188
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  4. 2022Neural Optimal Transportcited by 123
  5. 2017Large-Scale Optimal Transport and Mapping Estimationcited by 120
  6. 2018Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithmcited by 111
  7. 2014On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpointcited by 99
  8. 2019Wasserstein-2 Generative Networkscited by 96
  9. 2021Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmarkcited by 95
  10. 2020Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimizationcited by 89
  11. 2019Learning with minibatch Wasserstein : asymptotic and gradient propertiescited by 83
  12. 2018Optimal Control Via Neural Networks: A Convex Approachcited by 82
  13. 2021Generative Modeling with Optimal Transport Mapscited by 80
  14. 2022Supervised Training of Conditional Monge Mapscited by 60
  15. 2021Score-based Generative Neural Networks for Large-Scale Optimal Transportcited by 59
  16. 2021Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimizationcited by 50
  17. 2020Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networkscited by 49
  18. 2022Entropic Neural Optimal Transport via Diffusion Processescited by 49
  19. 2024Light and Optimal Schr"odinger Bridge Matchingcited by 45
  20. 2021Unbalanced Feature Transport for Exemplar-based Image Translationcited by 41
  21. 2021Neural Monge Map estimation and its applicationscited by 40
  22. 2023Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transportcited by 39
  23. 2022Neural Optimal Transport with General Cost Functionalscited by 34
  24. 2021Wasserstein GANs Work Because They Fail (to Approximate the Wasserstein Distance)cited by 33
  25. 20192-Wasserstein Approximation via Restricted Convex Potentials with Application to Improved Training for GANscited by 32
  26. 2023The Monge Gap: A Regularizer to Learn All Transport Mapscited by 32
  27. 2022Kernel Neural Optimal Transportcited by 31
  28. 2022On amortizing convex conjugates for optimal transportcited by 30
  29. 2023Energy-guided Entropic Neural Optimal Transportcited by 25
  30. 2022Wasserstein Iterative Networks for Barycenter Estimationcited by 23
  31. 2023Light Schr"odinger Bridgecited by 18
  32. 2021Input Convex Gradient Networkscited by 17
  33. 2024Estimating Barycenters of Distributions with Neural Optimal Transportcited by 14
  34. 2023Extremal Domain Translation with Neural Optimal Transportcited by 12
  35. 2023Principled Weight Initialisation for Input-Convex Neural Networkscited by 12
  36. 2022Numerical Methods for Large-Scale Optimal Transportcited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 8; 2025: 28; 2026: 38.

74 papers.

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By area:

  1. 2024Simplifying, Stabilizing and Scaling Continuous-Time Consistency Modelscited by 276
  2. 2024Consistency Flow Matching: Defining Straight Flows with Velocity Consistencycited by 101
  3. 2024PnP-Flow: Plug-and-Play Image Restoration with Flow Matchingcited by 92
  4. 2025One-Step Flow Policy Mirror Descentcited by 18
  5. 2025On the Relation between Rectified Flows and Optimal Transportcited by 15
  6. 2026Action-to-Action Flow Matchingcited by 15
  7. 2025Flow Matching Meets Biology and Life Science: A Surveycited by 14
  8. 2025Source-Guided Flow Matchingcited by 11
  9. 2025High-order Equivariant Flow Matching for Density Functional Theory Hamiltonian Predictioncited by 10
  10. 2024Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transportcited by 9
  11. 2026Mean Flow Policy with Instantaneous Velocity Constraint for One-step Action Generationcited by 9
  12. 2025Straight-Line Diffusion Model for Efficient 3D Molecular Generationcited by 8
38 more
  1. 2025Hierarchical Refinement: Optimal Transport to Infinity and Beyondcited by 8
  2. 2026Flow Matching is Adaptive to Manifold Structurescited by 8
  3. 2024Neural Entropycited by 7
  4. 2025OAT-FM: Optimal Acceleration Transport for Improved Flow Matchingcited by 7
  5. 2025Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generationcited by 7
  6. 2025Learning Straight Flows: Variational Flow Matching for Efficient Generationcited by 7
  7. 2025Learning of Population Dynamics: Inverse Optimization Meets JKO Schemecited by 6
  8. 2025AlignFlow: Improving Flow-based Generative Models with Semi-Discrete Optimal Transportcited by 6
  9. 2026Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generationcited by 5
  10. 2024NeuralPLexer3: Accurate Biomolecular Complex Structure Prediction with Flow Modelscited by 4
  11. 2025Improved Immiscible Diffusion: Accelerate Diffusion Training by Reducing Its Miscibilitycited by 4
  12. 2025Efficient Transferable Optimal Transport via Min-Sliced Transport Planscited by 4
  13. 2026FlowCast: Trajectory Forecasting for Scalable Zero-Cost Speculative Flow Matchingcited by 3
  14. 2026FastFlow: Accelerating The Generative Flow Matching Models with Bandit Inferencecited by 3
  15. 2026Geo2: Geometry-Guided Cross-view Geo-Localization and Image Synthesiscited by 3
  16. 2026Friction-Augmented Drifting Models for Resource-Efficient Domain Translationcited by 3
  17. 2026Tyche: One Step Flow for Efficient Probabilistic Weather Forecastingcited by 3
  18. 2024Speed-accuracy relations for diffusion models: Wisdom from nonequilibrium thermodynamics and optimal transportcited by 2
  19. 2025A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solverscited by 2
  20. 2025LVTINO: LAtent Video consisTency INverse sOlver for High Definition Video Restorationcited by 2
  21. 2025Deep Neural Networks Inspired by Differential Equationscited by 2
  22. 2025On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reductioncited by 2
  23. 2026Look-Ahead and Look-Back Flows: Training-Free Image Generation with Trajectory Smoothingcited by 2
  24. 2026Flow Matching from Viewpoint of Proximal Operatorscited by 2
  25. 2026COT-FM: Cluster-wise Optimal Transport Flow Matchingcited by 2
  26. 2026AssetGen: Deployable 3D Asset Generation at Interactive Speedcited by 2
  27. 2025Learning Straight Flows by Learning Curved Interpolantscited by 1
  28. 2025scSplit: Bringing Severity Cognizance to Image Decomposition in Fluorescence Microscopycited by 1
  29. 2025Weighted Conditional Flow Matchingcited by 1
  30. 2025Flow Matching Calibration for Simulation-Based Inference under Model Misspecificationcited by 1
  31. 2026Fast, faithful and photorealistic diffusion-based image super-resolution with enhanced Flow Map modelscited by 1
  32. 2026Variational Entropic Optimal Transportcited by 1
  33. 2026FAVE: Flow-based Average Velocity Establishment for Sequential Recommendationcited by 1
  34. 2026DirectTryOn: One-Step Virtual Try-On via Straightened Conditional Transportcited by 1
  35. 2026Optimal Transport Flow Matching by Designcited by 1
  36. 2026MedPCFM: Improving Medical Point Cloud Completion by Integrating Point Transformers and Flow Matchingcited by 1
  37. 2026Escaping the Euclidean Void: Manifold-Informed Flow Matching for Sequential Recommendationcited by 1
  38. 2024How to Spin an Object: First, Get the Shape Rightcited by 0

50 most cited of 74 All 74 (JSON)

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  1. 2026VTV-FM: Flow Matching through Variational Terminal-Velocity Closure
  2. 2025Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows
  3. 2026Path-independent Flow Matching for Multi-parameter Generative Dynamics
  4. 2025Beyond Optimal Transport: Model-Aligned Coupling for Flow Matching
  5. 2023Straighter Flow Matching via a Diffusion-Based Coupling Prior
  6. 2026Isokinetic Flow Matching for Pathwise Straightening of Generative Flows
  7. 2025Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching
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