# Optimal Flow Matching: Learning Straight Trajectories in Just One Step

arXiv: https://arxiv.org/abs/2403.13117 · HTML: https://mlsanity.com/paper/2403.13117/

## Key facts

- Cited by 78 papers in the map (data through 14 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.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 54)

- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [A Survey on Generative Diffusion Model](https://mlsanity.com/paper/2209.02646/) (2022)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Adversarial Latent Autoencoders](https://mlsanity.com/paper/2004.04467/) (2020)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm](https://mlsanity.com/paper/1802.04367/) (2018)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Learning with minibatch Wasserstein : asymptotic and gradient properties](https://mlsanity.com/paper/1910.04091/) (2019)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)

## Built on by (most cited first, 30 of 78)

- [Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models](https://mlsanity.com/paper/2410.11081/) (2024)
- [Consistency Flow Matching: Defining Straight Flows with Velocity Consistency](https://mlsanity.com/paper/2407.02398/) (2024)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [One-Step Flow Policy Mirror Descent](https://mlsanity.com/paper/2507.23675/) (2025)
- [On the Relation between Rectified Flows and Optimal Transport](https://mlsanity.com/paper/2505.19712/) (2025)
- [Action-to-Action Flow Matching](https://mlsanity.com/paper/2602.07322/) (2026)
- [Flow Matching Meets Biology and Life Science: A Survey](https://mlsanity.com/paper/2507.17731/) (2025)
- [Source-Guided Flow Matching](https://mlsanity.com/paper/2508.14807/) (2025)
- [High-order Equivariant Flow Matching for Density Functional Theory Hamiltonian Prediction](https://mlsanity.com/paper/2505.18817/) (2025)
- [Flow Matching is Adaptive to Manifold Structures](https://mlsanity.com/paper/2602.22486/) (2026)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Hierarchical Refinement: Optimal Transport to Infinity and Beyond](https://mlsanity.com/paper/2503.03025/) (2025)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Mean Flow Policy with Instantaneous Velocity Constraint for One-step Action Generation](https://mlsanity.com/paper/2602.13810/) (2026)
- [Straight-Line Diffusion Model for Efficient 3D Molecular Generation](https://mlsanity.com/paper/2503.02918/) (2025)
- [Learning Straight Flows: Variational Flow Matching for Efficient Generation](https://mlsanity.com/paper/2511.17583/) (2025)
- [Neural Entropy](https://mlsanity.com/paper/2409.03817/) (2024)
- [OAT-FM: Optimal Acceleration Transport for Improved Flow Matching](https://mlsanity.com/paper/2509.24936/) (2025)
- [Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generation](https://mlsanity.com/paper/2509.25157/) (2025)
- [AlignFlow: Improving Flow-based Generative Models with Semi-Discrete Optimal Transport](https://mlsanity.com/paper/2510.15038/) (2025)
- [Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generation](https://mlsanity.com/paper/2602.15022/) (2026)
- [NeuralPLexer3: Accurate Biomolecular Complex Structure Prediction with Flow Models](https://mlsanity.com/paper/2412.10743/) (2024)
- [Improved Immiscible Diffusion: Accelerate Diffusion Training by Reducing Its Miscibility](https://mlsanity.com/paper/2505.18521/) (2025)
- [Weighted Conditional Flow Matching](https://mlsanity.com/paper/2507.22270/) (2025)
- [Efficient Transferable Optimal Transport via Min-Sliced Transport Plans](https://mlsanity.com/paper/2511.19741/) (2025)
- [FlowCast: Trajectory Forecasting for Scalable Zero-Cost Speculative Flow Matching](https://mlsanity.com/paper/2602.01329/) (2026)
- [FastFlow: Accelerating The Generative Flow Matching Models with Bandit Inference](https://mlsanity.com/paper/2602.11105/) (2026)
- [On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reduction](https://mlsanity.com/paper/2512.18971/) (2025)
- [COT-FM: Cluster-wise Optimal Transport Flow Matching](https://mlsanity.com/paper/2603.13395/) (2026)
- [Geo2: Geometry-Guided Cross-view Geo-Localization and Image Synthesis](https://mlsanity.com/paper/2603.25819/) (2026)

## 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.