mlsanityData to 7 Oct 2026

Flow map matching with stochastic interpolants: A mathematical framework for consistency models

arXiv 2406.07507 ↗First posted 11 Jun 2024cs.LG math.DScited by 99

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

Abstract (arXiv)

Generative models based on dynamical equations such as flows and diffusions offer exceptional sample quality, but require computationally expensive numerical integration during inference. The advent of consistency models has enabled efficient one-step or few-step generation, yet despite their practical success, a systematic understanding of their design has been hindered by the lack of a comprehensive theoretical framework. Here we introduce Flow Map Matching (FMM), a principled framework for learning the two-time flow map of an underlying dynamical generative model, thereby providing this missing mathematical foundation. Leveraging stochastic interpolants, we propose training objectives both for distillation from a pre-trained velocity field and for direct training of a flow map over an interpolant or a forward diffusion process. Theoretically, we show that FMM unifies and extends a broad class of existing approaches for fast sampling, including consistency models, consistency trajectory models, and progressive distillation. Experiments on CIFAR-10 and ImageNet-32 highlight that our approach can achieve sample quality comparable to flow matching while reducing generation time by a factor of 10-20.

Key facts. Cited by 99 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 64 references that are in the map. First posted on arXiv 11 Jun 2024.

Builds on

54 of 64 references.

Closest ancestors · 4

Same direction, newest first

  1. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplings28 Apr
  2. 2023Improving and generalizing flow-based generative models with minibatch optimal transport1 Feb
  3. 2022Flow Matching for Generative Modeling6 Oct
  4. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow7 Sep

Other references · 50

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2021High-Resolution Image Synthesis with Latent Diffusion Modelscited by 18,694
  3. 2020Denoising Diffusion Implicit Modelscited by 9,655
  4. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  5. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,185
  6. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  7. 2022Generative Adversarial Networkscited by 5,603
  8. 2024Scaling Rectified Flow Transformers for High-Resolution Image Synthesiscited by 4,094
  9. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  10. 2021Improved Denoising Diffusion Probabilistic Modelscited by 3,736
  11. 2018Neural Ordinary Differential Equationscited by 3,733
  12. 2022Elucidating the Design Space of Diffusion-Based Generative Modelscited by 2,994
38 more
  1. 2023Diffusion Policy: Visuomotor Policy Learning via Action Diffusioncited by 2,730
  2. 2022Video Diffusion Modelscited by 2,280
  3. 2015Variational Inference with Normalizing Flowscited by 2,161
  4. 2016Density estimation using Real NVPcited by 2,029
  5. 2022DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Stepscited by 1,961
  6. 2023Consistency Modelscited by 1,778
  7. 2022Imagen Video: High Definition Video Generation with Diffusion Modelscited by 1,765
  8. 2023Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Modelscited by 1,465
  9. 2020Improved Techniques for Training Score-Based Generative Modelscited by 1,036
  10. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  11. 2022Tune-A-Video: One-Shot Tuning of Image Diffusion Models for Text-to-Video Generationcited by 958
  12. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  13. 2024SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformerscited by 707
  14. 2022Pseudo Numerical Methods for Diffusion Models on Manifoldscited by 695
  15. 2017Generative Adversarial Networks: An Overviewcited by 588
  16. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 584
  17. 2022Fast Sampling of Diffusion Models with Exponential Integratorcited by 526
  18. 2023Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusioncited by 448
  19. 2021Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speechcited by 445
  20. 2023Improved Techniques for Training Consistency Modelscited by 430
  21. 2021Knowledge Distillation in Iterative Generative Models for Improved Sampling Speedcited by 344
  22. 2021Gotta Go Fast When Generating Data with Score-Based Modelscited by 218
  23. 2016Unrolled Generative Adversarial Networkscited by 208
  24. 2022Learning Fast Samplers for Diffusion Models by Differentiating Through Sample Qualitycited by 181
  25. 2022Fast Sampling of Diffusion Models via Operator Learningcited by 173
  26. 2021Diff-TTS: A Denoising Diffusion Model for Text-to-Speechcited by 156
  27. 2022ProDiff: Progressive Fast Diffusion Model For High-Quality Text-to-Speechcited by 147
  28. 2022Conditional Diffusion Probabilistic Model for Speech Enhancementcited by 132
  29. 2021Learning to Efficiently Sample from Diffusion Probabilistic Modelscited by 129
  30. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Modelscited by 102
  31. 2023Stochastic interpolants with data-dependent couplingscited by 94
  32. 2024Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processescited by 61
  33. 2023Consistent Diffusion Models: Mitigating Sampling Drift by Learning to be Consistentcited by 58
  34. 2022Probability flow solution of the Fokker-Planck equationcited by 46
  35. 2022Self-Consistency of the Fokker-Planck Equationcited by 18
  36. 2024Bidirectional Consistency Modelscited by 18
  37. 2024Generalized Consistency Trajectory Models for Image Manipulationcited by 11
  38. 2023On the Equivalence of Consistency-Type Models: Consistency Models, Consistent Diffusion Models, and Fokker-Planck Regularizationcited by 10

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 24; 2026: 73.

99 papers.

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By area:

  1. 2026Generative Modeling via Driftingcited by 98
  2. 2024One-Step Diffusion Distillation through Score Implicit Matchingcited by 68
  3. 2026One-step Latent-free Image Generation with Pixel Mean Flowscited by 49
  4. 2025Memorization and Regularization in Generative Diffusion Modelscited by 39
  5. 2026Categorical Flow Mapscited by 37
  6. 2025Noise Hypernetworks: Amortizing Test-Time Compute in Diffusion Modelscited by 35
  7. 2025Much Ado About Noising: Dispelling the Myths of Generative Robotic Controlcited by 35
  8. 2025Transition Models: Rethinking the Generative Learning Objectivecited by 32
  9. 2026Meta Flow Maps enable scalable reward alignmentcited by 32
  10. 2026Diamond Maps: Efficient Reward Alignment via Stochastic Flow Mapscited by 26
  11. 2025pi-Flow: Policy-Based Few-Step Generation via Imitation Distillationcited by 25
  12. 2026Discrete Flow Mapscited by 23
38 more
  1. 2026AnyFlow: Any-Step Video Diffusion Model with On-Policy Flow Map Distillationcited by 23
  2. 2025Joint Distillation for Fast Likelihood Evaluation and Sampling in Flow-based Modelscited by 17
  3. 2026Reinforce Adjoint Matching: Scaling RL Post-Training of Diffusion and Flow-Matching Modelscited by 17
  4. 2025FACM: Flow-Anchored Consistency Modelscited by 16
  5. 2025Lipschitz-Guided Design of Interpolation Schedules in Generative Modelscited by 15
  6. 2025Ultra-Fast Language Generation via Discrete Diffusion Divergence Instructcited by 15
  7. 2025Flow Map Distillation Without Datacited by 13
  8. 2025Ideas in Inference-time Scaling can Benefit Generative Pre-training Algorithmscited by 12
  9. 2025Flow Matching with Semidiscrete Couplingscited by 12
  10. 2025FALCON: Few-step Accurate Likelihoods for Continuous Flowscited by 12
  11. 2026Salt: Self-Consistent Distribution Matching with Cache-Aware Training for Fast Video Generationcited by 11
  12. 2025DiffRatio: Training One-Step Diffusion Models Without Teacher Supervisioncited by 8
  13. 2026Posterior Refinement: Fast Language Generation via Any-Order Flow Mapscited by 8
  14. 2026Skywork UniPic 3.0: Unified Multi-Image Composition via Sequence Modelingcited by 7
  15. 2026Parallel Decoding Distillation for Fast Image and Video Generationcited by 7
  16. 2025AlignFlow: Improving Flow-based Generative Models with Semi-Discrete Optimal Transportcited by 6
  17. 2025MeanFlow Transformers with Representation Autoencoderscited by 6
  18. 2026Scaling Categorical Flow Mapscited by 6
  19. 2026Kernel-Gradient Drifting Modelscited by 6
  20. 2026Hacking Generative Perplexity: Why Unconditional Text Evaluation Needs Distributional Metricscited by 6
  21. 2025Functional Mean Flow in Hilbert Spacecited by 5
  22. 2026Learning Hamiltonian Flow Maps: Mean Flow Consistency for Large-Timestep Molecular Dynamicscited by 5
  23. 2026Riemannian MeanFlowcited by 5
  24. 2026How to Guide Your Flow: Few-Step Alignment via Flow Map Reward Guidancecited by 5
  25. 2026Flow Reasoning Models: Turning Flows Into Efficient Recurrent Reasonerscited by 5
  26. 2025Pathway to O(√d) Complexity bound under Wasserstein metric of flow-based modelscited by 4
  27. 2025On the Design of One-step Diffusion via Shortcutting Flow Pathscited by 4
  28. 2026Distribution Matching Distillation without Fake Score Networkcited by 4
  29. 2025Self-Evaluation Unlocks Any-Step Text-to-Image Generationcited by 3
  30. 2026Bi-Anchor Interpolation Solver for Accelerating Generative Modelingcited by 3
  31. 2026Riemannian MeanFlow for One-Step Generation on Manifoldscited by 3
  32. 2026Aligning Few-Step Generative Models by Amortizing Sample-based Variational Inferencecited by 3
  33. 2026Let It Be Simple: One-Step Action Generation for Vision-Language-Action Modelscited by 3
  34. 2026Flow-Map GRPO: Reinforcement Learning for Few-Step Flow-Map Generators via Anchored Stochastic Compositioncited by 3
  35. 2026Stabilizing Consistency Training: A Flow Map Analysis and Self-Distillationcited by 2
  36. 2026Trajectory Consistency for One-Step Generation on Euler Mean Flowscited by 2
  37. 2026Temporal Pair Consistency for Variance-Reduced Flow Matchingcited by 2
  38. 2026COT-FM: Cluster-wise Optimal Transport Flow Matchingcited by 2

50 most cited of 99 All 99 (JSON)

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  1. 2025How to build a consistency model: Learning flow maps via self-distillation
  2. 2026Path-independent Flow Matching for Multi-parameter Generative Dynamics
  3. 2024Local Flow Matching Generative Models
  4. 2025Diff2Flow: Training Flow Matching Models via Diffusion Model Alignment
  5. 2026Flow Matching is Adaptive to Manifold Structures
  6. 2026ODE-free Neural Flow Matching for One-Step Generative Modeling
  7. 2025Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching
  8. 2025Decoupled MeanFlow: Turning Flow Models into Flow Maps for Accelerated Sampling