mlsanityData to 14 Oct 2026

Error Bounds for Flow Matching Methods

arXiv 2305.16860 ↗First posted 26 May 2023stat.ML cs.LGcited by 91

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Score-based generative models are a popular class of generative modelling techniques relying on stochastic differential equations (SDE). From their inception, it was realized that it was also possible to perform generation using ordinary differential equations (ODE) rather than SDE. This led to the introduction of the probability flow ODE approach and denoising diffusion implicit models. Flow matching methods have recently further extended these ODE-based approaches and approximate a flow between two arbitrary probability distributions. Previous work derived bounds on the approximation error of diffusion models under the stochastic sampling regime, given assumptions on the L2 loss. We present error bounds for the flow matching procedure using fully deterministic sampling, assuming an L2 bound on the approximation error and a certain regularity condition on the data distributions.

Key facts. Cited by 91 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 31 of its 42 references that are in the map. First posted on arXiv 26 May 2023.

Builds on

31 of 42 references.

Closest ancestors · 16

Same direction, newest first

  1. 2023The probability flow ODE is provably fast19 May
  2. 2023Iterative α-(de)Blending: a Minimalist Deterministic Diffusion Model5 May
  3. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers6 Mar
  4. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  5. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  6. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  7. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Models31 Aug
  8. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  9. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  10. 2022Score-Based Generative Models Detect Manifolds2 Jun
  11. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  12. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
4 more
  1. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  3. 2020Denoising Diffusion Implicit Models6 Oct
  4. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 15

  1. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,585
  2. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,355
  3. 2022Flow Matching for Generative Modelingcited by 5,702
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  5. 2018Neural Ordinary Differential Equationscited by 3,779
  6. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,700
  7. 2015Variational Inference with Normalizing Flowscited by 2,170
  8. 2021Image Super-Resolution via Iterative Refinementcited by 1,511
  9. 2022Building Normalizing Flows with Stochastic Interpolantscited by 995
  10. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 939
  11. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 594
  12. 2021Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speechcited by 447
3 more
  1. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 119
  2. 2021Moser Flow: Divergence-based Generative Modeling on Manifoldscited by 67
  3. 2022Matching Normalizing Flows and Probability Paths on Manifoldscited by 57

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 7; 2024: 23; 2025: 36; 2026: 25.

91 papers.

Sort:

By area:

  1. 2024SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformerscited by 737
  2. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 235
  3. 2024PnP-Flow: Plug-and-Play Image Restoration with Flow Matchingcited by 97
  4. 2024Accelerating Convergence of Score-Based Diffusion Models, Provablycited by 76
  5. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Modelscited by 65
  6. 2024Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorialcited by 60
  7. 2023Learning Mixtures of Gaussians Using the DDPM Objectivecited by 57
  8. 2024Convergence Analysis of Probability Flow ODE for Score-based Generative Modelscited by 57
  9. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraintscited by 52
  10. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptionscited by 50
  11. 2024How Discrete and Continuous Diffusion Meet: Comprehensive Analysis of Discrete Diffusion Models via a Stochastic Integral Frameworkcited by 48
  12. 2024Critical windows: non-asymptotic theory for feature emergence in diffusion modelscited by 47
38 more
  1. 2024Learning general Gaussian mixtures with efficient score matchingcited by 44
  2. 2025Low-dimensional adaptation of diffusion models: Convergence in total variationcited by 39
  3. 2023Analysis of learning a flow-based generative model from limited sample complexitycited by 38
  4. 2024A Good Score Does not Lead to A Good Generative Modelcited by 36
  5. 2024Stochastic Runge-Kutta Methods: Provable Acceleration of Diffusion Modelscited by 35
  6. 2023Convergence of flow-based generative models via proximal gradient descent in Wasserstein spacecited by 34
  7. 2024Contractive Diffusion Probabilistic Modelscited by 29
  8. 2024Flow matching achieves almost minimax optimal convergencecited by 29
  9. 2024Unifying Generation and Prediction on Graphs with Latent Graph Diffusioncited by 25
  10. 2024Unraveling the Smoothness Properties of Diffusion Models: A Gaussian Mixture Perspectivecited by 24
  11. 2025Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distancecited by 21
  12. 2025Joint Velocity-Growth Flow Matching for Single-Cell Dynamics Modelingcited by 20
  13. 2024How Do Flow Matching Models Memorize and Generalize in Sample Data Subspaces?cited by 19
  14. 2025On the Relation between Rectified Flows and Optimal Transportcited by 18
  15. 2023Gaussian Interpolation Flowscited by 17
  16. 2024Exploring the Frontiers of Softmax: Provable Optimization, Applications in Diffusion Model, and Beyondcited by 15
  17. 2024Theoretical guarantees in KL for Diffusion Flow Matchingcited by 14
  18. 2024Unified Convergence Analysis for Score-Based Diffusion Models with Deterministic Samplerscited by 14
  19. 2025A solvable model of learning generative diffusion: theory and insightscited by 14
  20. 2025A Theoretical Analysis of Discrete Flow Matching Generative Modelscited by 14
  21. 2025Source-Guided Flow Matchingcited by 13
  22. 2023Closing the ODE-SDE gap in score-based diffusion models through the Fokker-Planck equationcited by 12
  23. 2025On the minimax optimality of Flow Matching through the connection to kernel density estimationcited by 10
  24. 2023Conditional Stochastic Interpolation for Generative Learningcited by 9
  25. 2025Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Modelscited by 9
  26. 2025DoubleGen: Debiased Generative Modeling of Counterfactualscited by 9
  27. 2025Error Analysis of Discrete Flow with Generator Matchingcited by 9
  28. 2026Better Source, Better Flow: Learning Condition-Dependent Source Distribution for Flow Matchingcited by 9
  29. 2024Path-Guided Particle-based Samplingcited by 8
  30. 2025Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Modelscited by 8
  31. 2025Efficient Regression-Based Training of Normalizing Flows for Boltzmann Generatorscited by 8
  32. 2025Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domainscited by 8
  33. 2026Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalitiescited by 8
  34. 2025Non-asymptotic error bounds for probability flow ODEs under weak log-concavitycited by 7
  35. 2026High-accuracy and dimension-free sampling with diffusionscited by 7
  36. 2025Distribution estimation via Flow Matching with Lipschitz guaranteescited by 6
  37. 2025On Flow Matching KL Divergencecited by 6
  38. 2024Local Flow Matching Generative Modelscited by 5

50 most cited of 91 All 91 (JSON)

Similar, not cited

Close in wording, no citation either way

  1. 2025Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows
  2. 2025Minimax Optimality of the Probability Flow ODE for Diffusion Models
  3. 2026A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models
  4. 2026Diffusion Flow Matching: Dimension-Improved KL Bounds and Wasserstein Guarantees
  5. 2023Exploring the Optimal Choice for Generative Processes in Diffusion Models: Ordinary vs Stochastic Differential Equations
  6. 2026From Euler to Dormand-Prince: ODE Solvers for Flow Matching Generative Models
  7. 2025An Introduction to Flow Matching and Diffusion Models
  8. 2026Probability flow ODEs in score-based and reflected diffusion models