mlsanityData to 7 Oct 2026

Convergence Analysis of Probability Flow ODE for Score-based Generative Models

arXiv 2404.09730 ↗First posted 15 Apr 2024cs.LG cs.NA math.CAcited by 56

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Score-based generative models have emerged as a powerful approach for sampling high-dimensional probability distributions. Despite their effectiveness, their theoretical underpinnings remain relatively underdeveloped. In this work, we study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives. Assuming access to L2-accurate estimates of the score function, we prove the total variation between the target and the generated data distributions can be bounded above by O(d3/4δ1/2) in the continuous time level, where d denotes the data dimension and δ represents the L2-score matching error. For practical implementations using a p-th order Runge-Kutta integrator with step size h, we establish error bounds of O(d3/4δ1/2 + d·(dh)p) at the discrete level. Finally, we present numerical studies on problems up to 128 dimensions to verify our theory.

Key facts. Cited by 56 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 53 references that are in the map. First posted on arXiv 15 Apr 2024.

Builds on

39 of 53 references.

Closest ancestors · 24

Same direction, newest first

  1. 2024Accelerating Convergence of Score-Based Diffusion Models, Provably6 Mar
  2. 2024Scaling Rectified Flow Transformers for High-Resolution Image Synthesis5 Mar
  3. 2024Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Models3 Mar
  4. 2024Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial12 Feb
  5. 2024Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances31 Jan
  6. 2024Contractive Diffusion Probabilistic Models23 Jan
  7. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models15 Jun
  8. 2023Error Bounds for Flow Matching Methods26 May
  9. 2023The probability flow ODE is provably fast19 May
  10. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers6 Mar
  11. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data14 Feb
  12. 2022Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance13 Dec
12 more
  1. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  2. 2022DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models2 Nov
  3. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  4. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  5. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  6. 2022DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps2 Jun
  7. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  8. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  9. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  10. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  11. 2020Denoising Diffusion Implicit Models6 Oct
  12. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 15

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,539
  3. 2022Flow Matching for Generative Modelingcited by 5,426
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  5. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,550
  6. 2015Variational Inference with Normalizing Flowscited by 2,161
  7. 2018On the Spectral Bias of Neural Networkscited by 1,321
  8. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  9. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  10. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  11. 2020Jukebox: A Generative Model for Musiccited by 535
  12. 2021Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speechcited by 445
3 more
  1. 2019Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networkscited by 360
  2. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 118
  3. 2022Probability flow solution of the Fokker-Planck equationcited by 46

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 17; 2025: 22; 2026: 17.

56 papers.

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  2. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptionscited by 48
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  4. 2024Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionalitycited by 44
  5. 2024Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexitycited by 42
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  8. 2024Provable Acceleration for Diffusion Models under Minimal Assumptionscited by 27
  9. 2025Minimax Optimality of the Probability Flow ODE for Diffusion Modelscited by 23
  10. 2025Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixturescited by 23
  11. 2025Quantitative Clustering in Mean-Field Transformer Modelscited by 23
  12. 2024Instance-dependent Convergence Theory for Diffusion Modelscited by 21
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  3. 2024Constrained Diffusion Models via Dual Trainingcited by 19
  4. 2025Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Accelerationcited by 17
  5. 2026Confidence-Based Decoding is Provably Efficient for Diffusion Language Modelscited by 17
  6. 2025Faster Diffusion Models via Higher-Order Approximationcited by 16
  7. 2025Optimal Convergence Analysis of DDPM for General Distributionscited by 16
  8. 2025On the Interpolation Effect of Score Smoothing in Diffusion Modelscited by 15
  9. 2024Unified Convergence Analysis for Score-Based Diffusion Models with Deterministic Samplerscited by 14
  10. 2026High-accuracy sampling for diffusion models and log-concave distributionscited by 13
  11. 2024Latent Diffusion Bridges for Unsupervised Musical Audio Timbre Transfercited by 9
  12. 2026Adaptation to Intrinsic Dependence in Diffusion Language Modelscited by 9
  13. 2025Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Modelscited by 8
  14. 2025A Sharp KL-Convergence Analysis for Diffusion Models under Minimal Assumptionscited by 8
  15. 2025Modular MeanFlow: Towards Stable and Scalable One-Step Generative Modelingcited by 8
  16. 2025Sublinear iterations can suffice even for DDPMscited by 7
  17. 2026High-accuracy and dimension-free sampling with diffusionscited by 7
  18. 2024Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplerscited by 6
  19. 2024Parallel Simulation for Log-concave Sampling and Score-based Diffusion Modelscited by 6
  20. 2024Global Well-posedness and Convergence Analysis of Score-based Generative Models via Sharp Lipschitz Estimatescited by 5
  21. 2024Posterior sampling via Langevin dynamics based on generative priorscited by 5
  22. 2024Local Flow Matching Generative Modelscited by 5
  23. 2025Score-based deterministic density samplingcited by 4
  24. 2025Convergence Of Consistency Model With Multistep Sampling Under General Data Assumptionscited by 4
  25. 2026Provable diffusion-based posterior sampling for linear inverse problems via DDIMcited by 4
  26. 2025A Non-Asymptotic Convergent Analysis for Scored-Based Graph Generative Model via a System of Stochastic Differential Equationscited by 3
  27. 2026ART for Diffusion Sampling: A Reinforcement Learning Approach to Timestep Schedulecited by 3
  28. 2025Regularization can make diffusion models more efficientcited by 2
  29. 2025Multi-Step Consistency Models: Fast Generation with Theoretical Guaranteescited by 2
  30. 2025Information Science Principles of Machine Learning: A Causal Chain Meta-Framework Based on Formalized Information Mappingcited by 2
  31. 2026Error Analysis of Bayesian Inverse Problems with Generative Priorscited by 2
  32. 2024On the Relation Between Linear Diffusion and Power Iterationcited by 1
  33. 2025Finite-Time Convergence Analysis of ODE-based Generative Models for Stochastic Interpolantscited by 1
  34. 2026Provably Reliable Classifier Guidance via Cross-Entropy Controlcited by 1
  35. 2026On the Robustness of Distribution Support under Diffusion Guidancecited by 1
  36. 2026Diffusion Models for Adaptive Sequential Data Generationcited by 1
  37. 2026Wasserstein Convergence of ODE-Based Samplers in Decentralized Diffusion Model via Velocity Field Decompositioncited by 1
  38. 2026Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choicescited by 1

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  2. 2023On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates
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  4. 2026Probability flow ODEs in score-based and reflected diffusion models
  5. 2026A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models
  6. 2025Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients
  7. 2024Discrete vs. Continuous Trade-offs for Generative Models
  8. 2023Geometry of Score Based Generative Models