mlsanityData to 7 Oct 2026

The probability flow ODE is provably fast

arXiv 2305.11798 ↗First posted 19 May 2023cs.LG math.ST stat.MLcited by 145

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

We provide the first polynomial-time convergence guarantees for the probability flow ODE implementation (together with a corrector step) of score-based generative modeling. Our analysis is carried out in the wake of recent results obtaining such guarantees for the SDE-based implementation (i.e., denoising diffusion probabilistic modeling or DDPM), but requires the development of novel techniques for studying deterministic dynamics without contractivity. Through the use of a specially chosen corrector step based on the underdamped Langevin diffusion, we obtain better dimension dependence than prior works on DDPM (O(√d) vs. O(d), assuming smoothness of the data distribution), highlighting potential advantages of the ODE framework.

Key facts. Cited by 145 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 39 of its 54 references that are in the map. First posted on arXiv 19 May 2023.

Builds on

39 of 54 references.

Closest ancestors · 21

Same direction, newest first

  1. 2023Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputation12 May
  2. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers6 Mar
  3. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  4. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  5. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  6. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Models31 Aug
  7. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  8. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  9. 2022Score-Based Generative Models Detect Manifolds2 Jun
  10. 2022DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps2 Jun
  11. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  12. 2022Score-Based Generative Models for Molecule Generation7 Mar
9 more
  1. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  2. 2021Structured Denoising Diffusion Models in Discrete State-Spaces7 Jul
  3. 2021Variational Diffusion Models1 Jul
  4. 2021Score-based Generative Modeling in Latent Space10 Jun
  5. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  6. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  7. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  8. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  9. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 18

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,539
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  4. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  5. 2022Diffusion Policies as an Expressive Policy Class for Offline Reinforcement Learningcited by 543
  6. 2021Solving Inverse Problems in Medical Imaging with Score-Based Generative Modelscited by 514
  7. 2021Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contractioncited by 306
  8. 2021Learning Gradient Fields for Molecular Conformation Generationcited by 162
  9. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 118
  10. 2020Structured Logconcave Sampling with a Restricted Gaussian Oraclecited by 85
  11. 2022Improved analysis for a proximal algorithm for samplingcited by 72
  12. 2022Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localizationcited by 48
6 more
  1. 2022Probability flow solution of the Fokker-Planck equationcited by 46
  2. 2023Improved dimension dependence of a proximal algorithm for samplingcited by 29
  3. 2021A Proximal Algorithm for Sampling from Non-smooth Potentialscited by 23
  4. 2016Auxiliary gradient-based sampling algorithmscited by 21
  5. 2022A Proximal Algorithm for Samplingcited by 21
  6. 2022Localization Schemes: A Framework for Proving Mixing Bounds for Markov Chainscited by 16

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 14; 2024: 49; 2025: 53; 2026: 29.

145 papers.

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  1. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 230
  2. 2023Error Bounds for Flow Matching Methodscited by 88
  3. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Modelscited by 84
  4. 2024Accelerating Convergence of Score-Based Diffusion Models, Provablycited by 74
  5. 2024Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstructioncited by 73
  6. 2024An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimizationcited by 72
  7. 2024Accelerating Diffusion Sampling with Optimized Time Stepscited by 67
  8. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Modelscited by 64
  9. 2024Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorialcited by 60
  10. 2024Gradient Guidance for Diffusion Models: An Optimization Perspectivecited by 59
  11. 2024Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theorycited by 58
  12. 2023Learning Mixtures of Gaussians Using the DDPM Objectivecited by 56
38 more
  1. 2024Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distancescited by 56
  2. 2024Convergence Analysis of Probability Flow ODE for Score-based Generative Modelscited by 56
  3. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptionscited by 48
  4. 2024How Discrete and Continuous Diffusion Meet: Comprehensive Analysis of Discrete Diffusion Models via a Stochastic Integral Frameworkcited by 48
  5. 2025Fast Solvers for Discrete Diffusion Models: Theory and Applications of High-Order Algorithmscited by 47
  6. 2024What does guidance do? A fine-grained analysis in a simple settingcited by 45
  7. 2024Convergence Analysis of Discrete Diffusion Model: Exact Implementation through Uniformizationcited by 44
  8. 2024Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionalitycited by 44
  9. 2024Learning general Gaussian mixtures with efficient score matchingcited by 43
  10. 2024Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexitycited by 42
  11. 2024Critical windows: non-asymptotic theory for feature emergence in diffusion modelscited by 41
  12. 2023Deep Networks as Denoising Algorithms: Sample-Efficient Learning of Diffusion Models in High-Dimensional Graphical Modelscited by 39
  13. 2024Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalizationcited by 39
  14. 2023Analysis of learning a flow-based generative model from limited sample complexitycited by 38
  15. 2025Low-dimensional adaptation of diffusion models: Convergence in total variationcited by 38
  16. 2024A Good Score Does not Lead to A Good Generative Modelcited by 36
  17. 2024Stochastic Runge-Kutta Methods: Provable Acceleration of Diffusion Modelscited by 35
  18. 2023Convergence of flow-based generative models via proximal gradient descent in Wasserstein spacecited by 34
  19. 2023On Error Propagation of Diffusion Modelscited by 30
  20. 2024Contractive Diffusion Probabilistic Modelscited by 29
  21. 2024Faster Diffusion Sampling with Randomized Midpoints: Sequential and Parallelcited by 29
  22. 2024Provable Acceleration for Diffusion Models under Minimal Assumptionscited by 27
  23. 2024Towards a mathematical theory for consistency training in diffusion modelscited by 25
  24. 2023On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimatescited by 24
  25. 2024Diffusion Model for Data-Driven Black-Box Optimizationcited by 24
  26. 2024Unraveling the Smoothness Properties of Diffusion Models: A Gaussian Mixture Perspectivecited by 24
  27. 2024Convergence of Score-Based Discrete Diffusion Models: A Discrete-Time Analysiscited by 24
  28. 2023Improved Convergence of Score-Based Diffusion Models via Prediction-Correctioncited by 23
  29. 2025Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixturescited by 23
  30. 2025Generalization through variance: how noise shapes inductive biases in diffusion modelscited by 23
  31. 2023Distribution learning via neural differential equations: a nonparametric statistical perspectivecited by 22
  32. 2024Instance-dependent Convergence Theory for Diffusion Modelscited by 21
  33. 2024A Survey on Statistical Theory of Deep Learning: Approximation, Training Dynamics, and Generative Modelscited by 20
  34. 2025Error Analyses of Auto-Regressive Video Diffusion Models: A Unified Frameworkcited by 20
  35. 2025Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distancecited by 20
  36. 2025Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approachcited by 18
  37. 2024U-Nets as Belief Propagation: Efficient Classification, Denoising, and Diffusion in Generative Hierarchical Modelscited by 17
  38. 2025Faster Diffusion Models via Higher-Order Approximationcited by 16

50 most cited of 145 All 145 (JSON)

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  6. 2025Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Models
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  8. 2025Diffusion Models: A Mathematical Introduction