mlsanityData to 14 Oct 2026

Simulation-free Schr"odinger bridges via score and flow matching

arXiv 2307.03672 ↗First posted 7 Jul 2023cs.LGcited by 83

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

We present simulation-free score and flow matching ([SF]2M), a simulation-free objective for inferring stochastic dynamics given unpaired samples drawn from arbitrary source and target distributions. Our method generalizes both the score-matching loss used in the training of diffusion models and the recently proposed flow matching loss used in the training of continuous normalizing flows. [SF]2M interprets continuous-time stochastic generative modeling as a Schr"odinger bridge problem. It relies on static entropy-regularized optimal transport, or a minibatch approximation, to efficiently learn the SB without simulating the learned stochastic process. We find that [SF]2M is more efficient and gives more accurate solutions to the SB problem than simulation-based methods from prior work. Finally, we apply [SF]2M to the problem of learning cell dynamics from snapshot data. Notably, [SF]2M is the first method to accurately model cell dynamics in high dimensions and can recover known gene regulatory networks from simulated data. Our code is available in the TorchCFM package at https://github.com/atong01/conditional-flow-matching.

Key facts. Cited by 83 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 55 of its 107 references that are in the map. First posted on arXiv 7 Jul 2023.

Builds on

55 of 107 references.

Closest ancestors · 12

Same direction, newest first

  1. 2023Diffusion Schr"odinger Bridge Matching29 Mar
  2. 2023I2SB: Image-to-Image Schr"odinger Bridge12 Feb
  3. 2022Applying Regularized Schr"odinger-Bridge-Based Stochastic Process in Generative Modeling15 Aug
  4. 2022Neural Lagrangian Schr"odinger Bridge: Diffusion Modeling for Population Dynamics11 Apr
  5. 2022Conditional Simulation Using Diffusion Schr"odinger Bridges27 Feb
  6. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  7. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  8. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  9. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  10. 2020Denoising Diffusion Implicit Models6 Oct
  11. 2020Denoising Diffusion Probabilistic Models19 Jun
  12. 2020Improved Techniques for Training Score-Based Generative Models16 Jun

Other references · 43

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  2. 2022Flow Matching for Generative Modelingcited by 5,702
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  4. 2018Neural Ordinary Differential Equationscited by 3,779
  5. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,700
  6. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,316
  7. 2022Building Normalizing Flows with Stochastic Interpolantscited by 995
  8. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 939
  9. 2023Improving and generalizing flow-based generative models with minibatch optimal transportcited by 879
  10. 2017Self-Normalizing Neural Networkscited by 598
  11. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 594
  12. 2017Learning Generative Models with Sinkhorn Divergencescited by 314
31 more
  1. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 292
  2. 2022Rectified Flow: A Marginal Preserving Approach to Optimal Transportcited by 283
  3. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplingscited by 278
  4. 2020How to train your neural ODE: the world of Jacobian and kinetic regularizationcited by 228
  5. 2017Near-linear time approximation algorithms for optimal transport via Sinkhorn iterationcited by 223
  6. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 219
  7. 2018DeepJDOT: Deep Joint Distribution Optimal Transport for Unsupervised Domain Adaptationcited by 214
  8. 2016Stochastic Optimization for Large-scale Optimal Transportcited by 189
  9. 2020TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamicscited by 160
  10. 2021Solving Schr"odinger Bridges via Maximum Likelihoodcited by 141
  11. 2021Unbalanced minibatch Optimal Transport; applications to Domain Adaptationcited by 125
  12. 2017Large-Scale Optimal Transport and Mapping Estimationcited by 123
  13. 2018Improving GANs Using Optimal Transportcited by 123
  14. 2021Deep Generative Learning via Schr"odinger Bridgecited by 104
  15. 2022Action Matching: Learning Stochastic Dynamics from Samplescited by 98
  16. 2021Proximal Optimal Transport Modeling of Population Dynamicscited by 93
  17. 2019Learning with minibatch Wasserstein : asymptotic and gradient propertiescited by 87
  18. 2019Statistical bounds for entropic optimal transport: sample complexity and the central limit theoremcited by 85
  19. 2023Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modelingcited by 80
  20. 2021Moser Flow: Divergence-based Generative Modeling on Manifoldscited by 67
  21. 2021Minibatch optimal transport distances; analysis and applicationscited by 65
  22. 2022Manifold Interpolating Optimal-Transport Flows for Trajectory Inferencecited by 64
  23. 2022The Schr"odinger Bridge between Gaussian Measures has a Closed Formcited by 63
  24. 2022Matching Normalizing Flows and Probability Paths on Manifoldscited by 57
  25. 2022Deep Generalized Schr"odinger Bridgecited by 47
  26. 2021Neural graphical modelling in continuous-time: consistency guarantees and algorithmscited by 42
  27. 2023Aligned Diffusion Schr"odinger Bridgescited by 38
  28. 2020Entropy-Regularized 2-Wasserstein Distance between Gaussian Measurescited by 32
  29. 2021Towards a mathematical theory of trajectory inferencecited by 31
  30. 2021Beyond Predictions in Neural ODEs: Identification and Interventionscited by 15
  31. 2022Sparsity in Continuous-Depth Neural Networkscited by 10

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 10; 2024: 16; 2025: 28; 2026: 29.

83 papers.

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By area:

  1. 2023Denoising Diffusion Bridge Modelscited by 178
  2. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generationcited by 150
  3. 2024ZigMa: A DiT-style Zigzag Mamba Diffusion Modelcited by 84
  4. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 77
  5. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 46
  6. 2024Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifoldcited by 43
  7. 2023Image Restoration Through Generalized Ornstein-Uhlenbeck Bridgecited by 41
  8. 2024Self-Consuming Generative Models with Curated Data Provably Optimize Human Preferencescited by 39
  9. 2023Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translationcited by 33
  10. 2023Harmonic Self-Conditioned Flow Matching for Multi-Ligand Docking and Binding Site Designcited by 30
  11. 2023Schrodinger Bridges Beat Diffusion Models on Text-to-Speech Synthesiscited by 30
  12. 2023Generative Modeling with Phase Stochastic Bridgescited by 25
38 more
  1. 2025VITA: Vision-to-Action Flow Matching Policycited by 25
  2. 2025Joint Velocity-Growth Flow Matching for Single-Cell Dynamics Modelingcited by 20
  3. 2025Curly Flow Matching for Learning Non-gradient Field Dynamicscited by 20
  4. 2024ImageFlowNet: Forecasting Multiscale Image-Level Trajectories of Disease Progression with Irregularly-Sampled Longitudinal Medical Imagescited by 19
  5. 2025h-Edit: Effective and Flexible Diffusion-Based Editing via Doob's h-Transformcited by 19
  6. 2023Light Schr"odinger Bridgecited by 18
  7. 2025DiverseFlow: Sample-Efficient Diverse Mode Coverage in Flowscited by 17
  8. 2024Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifoldscited by 13
  9. 2025FALCON: Few-step Accurate Likelihoods for Continuous Flowscited by 13
  10. 2023Energy-Guided Continuous Entropic Barycenter Estimation for General Costscited by 12
  11. 2024Efficient Trajectory Forecasting and Generation with Conditional Flow Matchingcited by 12
  12. 2024FlowNav: Combining Flow Matching and Depth Priors for Efficient Navigationcited by 12
  13. 2025Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridgecited by 12
  14. 2024Extended Flow Matching: a Method of Conditional Generation with Generalized Continuity Equationcited by 11
  15. 2025Variational Regularized Unbalanced Optimal Transport: Single Network, Least Actioncited by 11
  16. 2025Statistical Properties of Rectified Flowcited by 11
  17. 2025Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysiscited by 9
  18. 2026Better Source, Better Flow: Learning Condition-Dependent Source Distribution for Flow Matchingcited by 9
  19. 2025InfoBridge: Mutual Information estimation via Bridge Matchingcited by 7
  20. 2025Modeling Microenvironment Trajectories on Spatial Transcriptomics with NicheFlowcited by 7
  21. 2024Explicit Flow Matching: On The Theory of Flow Matching Algorithms with Applicationscited by 6
  22. 2025Adaptive Destruction Processes for Diffusion Samplerscited by 6
  23. 2025Oh SnapMMD! Forecasting Stochastic Dynamics Beyond the Schr"odinger Bridge's Endcited by 5
  24. 2025Branched Schr"odinger Bridge Matchingcited by 5
  25. 2025Neon: Negative Extrapolation From Self-Training Improves Image Generationcited by 5
  26. 2025LeapFactual: Reliable Visual Counterfactual Explanation Using Conditional Flow Matchingcited by 5
  27. 2024Conditional Lagrangian Wasserstein Flow for Time Series Imputationcited by 4
  28. 2025When and how can inexact generative models still sample from the data manifold?cited by 4
  29. 2026WarmPrior: Straightening Flow-Matching Policies with Temporal Priorscited by 4
  30. 2024Modeling Stochastic Conditional Dynamics from Sparse Observations via Kernel-Stabilized Flow Matchingcited by 3
  31. 2024Stable Autonomous Flow Matchingcited by 2
  32. 2025Variational Online Mirror Descent for Robust Learning in Schr"odinger Bridgecited by 2
  33. 2025Grasp2Grasp: Vision-Based Dexterous Grasp Translation via Schr"odinger Bridgescited by 2
  34. 2025Schr"odinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentrescited by 2
  35. 2026Learning Flow Distributions via Projection-Constrained Diffusion on Manifoldscited by 2
  36. 2026Distribution-Conditioned Transportcited by 2
  37. 2026Learning Lineage-guided Geodesics with Finsler Geometrycited by 2
  38. 2026Measure-to-measure Regression with Transformerscited by 2

50 most cited of 83 All 83 (JSON)

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  2. 2026Efficient Generative Modeling beyond Memoryless Diffusion via Adjoint Schr"odinger Bridge Matching
  3. 2025Data-to-Energy Stochastic Dynamics
  4. 2025Momentum Multi-Marginal Schr"odinger Bridge Matching
  5. 2025A Closed-Form Framework for Schr"odinger Bridges Between Arbitrary Densities
  6. 2023Generalized Schr"odinger Bridge Matching
  7. 2026Reflected Schr"odinger Bridge Matching
  8. 2024Schr"odinger Bridge Flow for Unpaired Data Translation