mlsanityData to 7 Oct 2026

Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

arXiv 2303.08797 ↗First posted 15 Mar 2023cs.LG cond-mat.dis-nn math.PRcited by 897

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Cited by 510 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo and Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called stochastic interpolants to bridge any two probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent density function of the interpolant is shown to satisfy a transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also construct estimators for the likelihood and the cross entropy of interpolant-based generative models, and we discuss connections with other methods such as score-based diffusion models, stochastic localization, probabilistic denoising, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.

Key facts. Cited by 897 papers in the map (data through 7 Oct 2026). Builds on 39 of its 57 references that are in the map. First posted on arXiv 15 Mar 2023.

Builds on

39 of 57 references.

References · 39

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  4. 2022Flow Matching for Generative Modelingcited by 5,426
  5. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  6. 2018Neural Ordinary Differential Equationscited by 3,733
  7. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,550
  8. 2022Elucidating the Design Space of Diffusion-Based Generative Modelscited by 2,994
  9. 2015Variational Inference with Normalizing Flowscited by 2,161
  10. 2016Density estimation using Real NVPcited by 2,029
  11. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  12. 2017Masked Autoregressive Flow for Density Estimationcited by 680
27 more
  1. 2021Maximum Likelihood Training of Score-Based Diffusion Modelscited by 670
  2. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modelingcited by 594
  3. 2019Neural Spline Flowscited by 590
  4. 2021Tackling the Generative Learning Trilemma with Denoising Diffusion GANscited by 515
  5. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptionscited by 440
  6. 2023Simple diffusion: End-to-end diffusion for high resolution imagescited by 413
  7. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 286
  8. 2022Rectified Flow: A Marginal Preserving Approach to Optimal Transportcited by 277
  9. 2021Score-Based Generative Modeling with Critically-Damped Langevin Diffusioncited by 230
  10. 2020How to train your neural ODE: the world of Jacobian and kinetic regularizationcited by 228
  11. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 216
  12. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptionscited by 215
  13. 2022Dual Diffusion Implicit Bridges for Image-to-Image Translationcited by 212
  14. 2018Neural Autoregressive Flowscited by 207
  15. 2022Convergence of score-based generative modeling for general data distributionscited by 179
  16. 2022Convergence for score-based generative modeling with polynomial complexitycited by 176
  17. 2020TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamicscited by 155
  18. 2020OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transportcited by 143
  19. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Modelscited by 102
  20. 2020Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimizationcited by 89
  21. 2020Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimationcited by 87
  22. 2022Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matchingcited by 79
  23. 2022Matching Normalizing Flows and Probability Paths on Manifoldscited by 57
  24. 2022Probability flow solution of the Fokker-Planck equationcited by 46
  25. 2023Aligned Diffusion Schr"odinger Bridgescited by 38
  26. 2018Monge-Amp`ere Flow for Generative Modelingcited by 33
  27. 2019Potential Flow Generator with L2 Optimal Transport Regularity for Generative Modelscited by 31

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 45; 2024: 149; 2025: 322; 2026: 381.

897 papers.

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  5. 2023InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generationcited by 387
  6. 2024Simplifying, Stabilizing and Scaling Continuous-Time Consistency Modelscited by 276
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  8. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 230
  9. 2024Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Controlcited by 222
  10. 2024A Survey on Diffusion Models for Inverse Problemscited by 173
  11. 2023Denoising Diffusion Bridge Modelscited by 171
  12. 2025MixGRPO: Unlocking Flow-based GRPO Efficiency with Mixed ODE-SDEcited by 168
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  4. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generationcited by 147
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  6. 2024Lumina-T2X: Transforming Text into Any Modality, Resolution, and Duration via Flow-based Large Diffusion Transformerscited by 137
  7. 2024Dynamical Regimes of Diffusion Modelscited by 133
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  10. 2024AlphaFold Meets Flow Matching for Generating Protein Ensemblescited by 105
  11. 2023DiffiT: Diffusion Vision Transformers for Image Generationcited by 104
  12. 2025Improved Mean Flows: On the Challenges of Fastforward Generative Modelscited by 102
  13. 2024Flow map matching with stochastic interpolants: A mathematical framework for consistency modelscited by 99
  14. 2026Generative Modeling via Driftingcited by 98
  15. 2023Flow Matching for Scalable Simulation-Based Inferencecited by 97
  16. 2025How to build a consistency model: Learning flow maps via self-distillationcited by 96
  17. 2024PeRFlow: Piecewise Rectified Flow as Universal Plug-and-Play Acceleratorcited by 95
  18. 2023Stochastic interpolants with data-dependent couplingscited by 94
  19. 2024PnP-Flow: Plug-and-Play Image Restoration with Flow Matchingcited by 92
  20. 2024LN3DIFF++: Scalable Latent Neural Fields Diffusion for Speedy 3D Generationcited by 91
  21. 2024Align Your Steps: Optimizing Sampling Schedules in Diffusion Modelscited by 91
  22. 2023Error Bounds for Flow Matching Methodscited by 88
  23. 2024ZigMa: A DiT-style Zigzag Mamba Diffusion Modelcited by 84
  24. 2023Simulation-free Schr"odinger bridges via score and flow matchingcited by 81
  25. 2025Align Your Flow: Scaling Continuous-Time Flow Map Distillationcited by 81
  26. 2024Variational Flow Matching for Graph Generationcited by 78
  27. 2025Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matchingcited by 78
  28. 2025Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Trainingcited by 78
  29. 2024An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimizationcited by 72
  30. 2024All-in-one simulation-based inferencecited by 71
  31. 2024NETS: A Non-Equilibrium Transport Samplercited by 67
  32. 2025Proteina: Scaling Flow-based Protein Structure Generative Modelscited by 63
  33. 2024Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processescited by 61
  34. 2025EditScore: Unlocking Online RL for Image Editing via High-Fidelity Reward Modelingcited by 61
  35. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensionscited by 60
  36. 2023On the Design Fundamentals of Diffusion Models: A Surveycited by 59
  37. 2024Gradient Guidance for Diffusion Models: An Optimization Perspectivecited by 59
  38. 2024Rectified Diffusion: Straightness Is Not Your Need in Rectified Flowcited by 59

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  4. 2024Speed-accuracy relations for diffusion models: Wisdom from nonequilibrium thermodynamics and optimal transport
  5. 2023Non-Denoising Forward-Time Diffusions
  6. 2026Probability flow ODEs in score-based and reflected diffusion models
  7. 2023FreeFlow: A Comprehensive Understanding on Diffusion Probabilistic Models via Optimal Transport
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