mlsanityData to 7 Oct 2026

Building Normalizing Flows with Stochastic Interpolants

arXiv 2209.15571 ↗First posted 30 Sep 2022cs.LG stat.MLcited by 964

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Cited by 554 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

A generative model based on a continuous-time normalizing flow between any pair of base and target probability densities is proposed. The velocity field of this flow is inferred from the probability current of a time-dependent density that interpolates between the base and the target in finite time. Unlike conventional normalizing flow inference methods based the maximum likelihood principle, which require costly backpropagation through ODE solvers, our interpolant approach leads to a simple quadratic loss for the velocity itself which is expressed in terms of expectations that are readily amenable to empirical estimation. The flow can be used to generate samples from either the base or target, and to estimate the likelihood at any time along the interpolant. In addition, the flow can be optimized to minimize the path length of the interpolant density, thereby paving the way for building optimal transport maps. In situations where the base is a Gaussian density, we also show that the velocity of our normalizing flow can also be used to construct a diffusion model to sample the target as well as estimate its score. However, our approach shows that we can bypass this diffusion completely and work at the level of the probability flow with greater simplicity, opening an avenue for methods based solely on ordinary differential equations as an alternative to those based on stochastic differential equations. Benchmarking on density estimation tasks illustrates that the learned flow can match and surpass conventional continuous flows at a fraction of the cost, and compares well with diffusions on image generation on CIFAR-10 and ImageNet 32×32. The method scales ab-initio ODE flows to previously unreachable image resolutions, demonstrated up to 128×128.

Key facts. Cited by 964 papers in the map (data through 7 Oct 2026). Builds on 38 of its 61 references that are in the map. First posted on arXiv 30 Sep 2022.

Builds on

38 of 61 references.

References · 38

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  3. 2021High-Resolution Image Synthesis with Latent Diffusion Modelscited by 18,694
  4. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  5. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  6. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,539
  7. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,322
  8. 2021Improved Denoising Diffusion Probabilistic Modelscited by 3,736
  9. 2018Neural Ordinary Differential Equationscited by 3,733
  10. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,550
  11. 2018Large Scale GAN Training for High Fidelity Natural Image Synthesiscited by 2,609
  12. 2015Variational Inference with Normalizing Flowscited by 2,161
26 more
  1. 2016Density estimation using Real NVPcited by 2,029
  2. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  3. 2019Generating Diverse High-Fidelity Images with VQ-VAE-2cited by 1,376
  4. 2015Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)cited by 1,061
  5. 2016Pixel Recurrent Neural Networkscited by 814
  6. 2022Autoregressive Image Generation using Residual Quantizationcited by 706
  7. 2017Masked Autoregressive Flow for Density Estimationcited by 680
  8. 2021Maximum Likelihood Training of Score-Based Diffusion Modelscited by 670
  9. 2022Equivariant Diffusion for Molecule Generation in 3Dcited by 666
  10. 2019Neural Spline Flowscited by 590
  11. 2021Tackling the Generative Learning Trilemma with Denoising Diffusion GANscited by 515
  12. 2015MADE: Masked Autoencoder for Distribution Estimationcited by 306
  13. 2022Rectified Flow: A Marginal Preserving Approach to Optimal Transportcited by 277
  14. 2022Cold Diffusion: Inverting Arbitrary Image Transforms Without Noisecited by 275
  15. 2020How to train your neural ODE: the world of Jacobian and kinetic regularizationcited by 228
  16. 2018Neural Autoregressive Flowscited by 207
  17. 2020OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transportcited by 143
  18. 2021Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimationcited by 95
  19. 2020Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimizationcited by 89
  20. 2022Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matchingcited by 79
  21. 2018Generating High Fidelity Images with Subscale Pixel Networks and Multidimensional Upscalingcited by 74
  22. 2021Moser Flow: Divergence-based Generative Modeling on Manifoldscited by 67
  23. 2021Density Ratio Estimation via Infinitesimal Classificationcited by 59
  24. 2022Matching Normalizing Flows and Probability Paths on Manifoldscited by 57
  25. 2021Simulating Diffusion Bridges with Score Matchingcited by 51
  26. 2022Probability flow solution of the Fokker-Planck equationcited by 46

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 6; 2023: 66; 2024: 146; 2025: 352; 2026: 394.

964 papers.

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  1. 2022Flow Matching for Generative Modelingcited by 5,426
  2. 2025FLUX.1 Kontext: Flow Matching for In-Context Image Generation and Editing in Latent Spacecited by 960
  3. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  4. 2023Improving and generalizing flow-based generative models with minibatch optimal transportcited by 835
  5. 2024Structured 3D Latents for Scalable and Versatile 3D Generationcited by 742
  6. 2024SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformerscited by 707
  7. 2024Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Thinkcited by 666
  8. 2025Mean Flows for One-step Generative Modelingcited by 524
  9. 2025DanceGRPO: Unleashing GRPO on Visual Generationcited by 388
  10. 2023InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generationcited by 387
  11. 2024Pyramidal Flow Matching for Efficient Video Generative Modelingcited by 354
  12. 2024Discrete Flow Matchingcited by 301
38 more
  1. 2024Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Designcited by 294
  2. 2024Simplifying, Stabilizing and Scaling Continuous-Time Consistency Modelscited by 276
  3. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplingscited by 267
  4. 2025MAGI-1: Autoregressive Video Generation at Scalecited by 262
  5. 2023Flow Matching on General Geometriescited by 260
  6. 2025OmniGen2: Towards Instruction-Aligned Multimodal Generationcited by 241
  7. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 230
  8. 2024FlowEdit: Inversion-Free Text-Based Editing Using Pre-Trained Flow Modelscited by 214
  9. 2024A Survey on Diffusion Models for Inverse Problemscited by 173
  10. 2023Denoising Diffusion Bridge Modelscited by 171
  11. 2023Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restorationcited by 164
  12. 2024Semantic Image Inversion and Editing using Rectified Stochastic Differential Equationscited by 162
  13. 2024JanusFlow: Harmonizing Autoregression and Rectified Flow for Unified Multimodal Understanding and Generationcited by 162
  14. 2025Flow Q-Learningcited by 162
  15. 2023Flow Matching in Latent Spacecited by 155
  16. 2023Diffusion Schr"odinger Bridge Matchingcited by 151
  17. 2025Steering Your Diffusion Policy with Latent Space Reinforcement Learningcited by 149
  18. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generationcited by 147
  19. 2023Matcha-TTS: A fast TTS architecture with conditional flow matchingcited by 142
  20. 2024Lumina-T2X: Transforming Text into Any Modality, Resolution, and Duration via Flow-based Large Diffusion Transformerscited by 137
  21. 2023Equivariant flow matchingcited by 131
  22. 2024Unlocking Guidance for Discrete State-Space Diffusion and Flow Modelscited by 124
  23. 2025Long-Context Autoregressive Video Modeling with Next-Frame Predictioncited by 119
  24. 2024ReNO: Enhancing One-step Text-to-Image Models through Reward-based Noise Optimizationcited by 115
  25. 2025Epona: Autoregressive Diffusion World Model for Autonomous Drivingcited by 115
  26. 2024Iterated Denoising Energy Matching for Sampling from Boltzmann Densitiescited by 113
  27. 2024Stable Flow: Vital Layers for Training-Free Image Editingcited by 111
  28. 2024Dirichlet Flow Matching with Applications to DNA Sequence Designcited by 110
  29. 2025Inductive Moment Matchingcited by 107
  30. 2024Improving the Training of Rectified Flowscited by 106
  31. 2024AlphaFold Meets Flow Matching for Generating Protein Ensemblescited by 105
  32. 2023DiffiT: Diffusion Vision Transformers for Image Generationcited by 104
  33. 2025Improved Mean Flows: On the Challenges of Fastforward Generative Modelscited by 102
  34. 2024Consistency Flow Matching: Defining Straight Flows with Velocity Consistencycited by 101
  35. 2023Guided Flows for Generative Modeling and Decision Makingcited by 99
  36. 2024Flow map matching with stochastic interpolants: A mathematical framework for consistency modelscited by 99
  37. 2023Flow Matching for Scalable Simulation-Based Inferencecited by 97
  38. 2025How to build a consistency model: Learning flow maps via self-distillationcited by 96

50 most cited of 964 All 964 (JSON)

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  1. 2026Fast Sampling for Flows and Diffusions with Lazy and Point Mass Stochastic Interpolants
  2. 2025Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching
  3. 2026A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models
  4. 2025Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows
  5. 2025Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans
  6. 2021Diffusion Normalizing Flow
  7. 2024Markovian Flow Matching: Accelerating MCMC with Continuous Normalizing Flows
  8. 2025The Principles of Diffusion Models