mlsanityData to 7 Oct 2026

Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial

arXiv 2402.07487 ↗First posted 12 Feb 2024cs.LG math.HOcited by 60

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

This is an expository article on the score-based diffusion models, with a particular focus on the formulation via stochastic differential equations (SDE). After a gentle introduction, we discuss the two pillars in the diffusion modeling -- sampling and score matching, which encompass the SDE/ODE sampling, score matching efficiency, the consistency models, and reinforcement learning. Short proofs are given to illustrate the main idea of the stated results. The article is primarily a technical introduction to the field, and practitioners may also find some analysis useful in designing new models or algorithms.

Key facts. Cited by 60 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 55 of its 113 references that are in the map. First posted on arXiv 12 Feb 2024.

Builds on

55 of 113 references.

Closest ancestors · 39

Same direction, newest first

  1. 2024Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances31 Jan
  2. 2024Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization28 Jan
  3. 2024Contractive Diffusion Probabilistic Models23 Jan
  4. 2023Wasserstein Convergence Guarantees for a General Class of Score-Based Generative Models18 Nov
  5. 2023Improved Techniques for Training Consistency Models22 Oct
  6. 2023Latent Consistency Models: Synthesizing High-Resolution Images with Few-Step Inference6 Oct
  7. 2023Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion1 Oct
  8. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization7 Aug
  9. 2023Improved Order Analysis and Design of Exponential Integrator for Diffusion Models Sampling4 Aug
  10. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models15 Jun
  11. 2023Error Bounds for Flow Matching Methods26 May
  12. 2023The probability flow ODE is provably fast19 May
27 more
  1. 2023Diffusion Models for Constrained Domains11 Apr
  2. 2023Reflected Diffusion Models10 Apr
  3. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers6 Mar
  4. 2023Consistency Models2 Mar
  5. 2023Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation1 Mar
  6. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data14 Feb
  7. 2023Optimizing DDPM Sampling with Shortcut Fine-Tuning31 Jan
  8. 2022Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance13 Dec
  9. 2022Categorical SDEs with Simplex Diffusion26 Oct
  10. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  11. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  12. 2022Diffusion Models: A Comprehensive Survey of Methods and Applications2 Sep
  13. 2022Understanding Diffusion Models: A Unified Perspective25 Aug
  14. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  15. 2022Classifier-Free Diffusion Guidance26 Jul
  16. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  17. 2022Score-Based Generative Models Detect Manifolds2 Jun
  18. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  19. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  20. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  21. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matching5 Jun
  22. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  23. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  24. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  25. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  26. 2020Denoising Diffusion Implicit Models6 Oct
  27. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 16

  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. 2022Make-A-Video: Text-to-Video Generation without Text-Video Datacited by 1,725
  7. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  8. 2023Training Diffusion Models with Reinforcement Learningcited by 939
  9. 2023Aligning Text-to-Image Models using Human Feedbackcited by 418
  10. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 381
  11. 2018Pros and Cons of GAN Evaluation Measurescited by 224
  12. 2021Policy Gradient and Actor-Critic Learning in Continuous Time and Space: Theory and Algorithmscited by 76
4 more
  1. 2022q-Learning in Continuous Timecited by 68
  2. 2022Statistical Efficiency of Score Matching: The View from Isoperimetrycited by 63
  3. 2023Policy Optimization for Continuous Reinforcement Learningcited by 30
  4. 2023Provable benefits of score matchingcited by 18

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 21; 2025: 24; 2026: 15.

60 papers.

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  1. 2024Understanding Reinforcement Learning-Based Fine-Tuning of Diffusion Models: A Tutorial and Reviewcited by 92
  2. 2025Inference-Time Alignment in Diffusion Models with Reward-Guided Generation: Tutorial and Reviewcited by 82
  3. 2024Accelerating Convergence of Score-Based Diffusion Models, Provablycited by 74
  4. 2024Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstructioncited by 73
  5. 2024An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimizationcited by 72
  6. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Modelscited by 64
  7. 2024Convergence Analysis of Probability Flow ODE for Score-based Generative Modelscited by 56
  8. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptionscited by 48
  9. 2024Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionalitycited by 44
  10. 2024Fine-tuning of diffusion models via stochastic control: entropy regularization and beyondcited by 42
  11. 2024Learning Mixtures of Gaussians Using Diffusion Modelscited by 41
  12. 2024On Statistical Rates and Provably Efficient Criteria of Latent Diffusion Transformers (DiTs)cited by 39
38 more
  1. 2025Low-dimensional adaptation of diffusion models: Convergence in total variationcited by 38
  2. 2024Stochastic Runge-Kutta Methods: Provable Acceleration of Diffusion Modelscited by 35
  3. 2025Score as Action: Fine-Tuning Diffusion Generative Models by Continuous-time Reinforcement Learningcited by 33
  4. 2024From optimal score matching to optimal samplingcited by 27
  5. 2024Provable Acceleration for Diffusion Models under Minimal Assumptionscited by 27
  6. 2024Preference Tuning with Human Feedback on Language, Speech, and Vision Tasks: A Surveycited by 23
  7. 2025Minimax Optimality of the Probability Flow ODE for Diffusion Modelscited by 23
  8. 2025Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixturescited by 23
  9. 2024Instance-dependent Convergence Theory for Diffusion Modelscited by 21
  10. 2024On Statistical Rates of Conditional Diffusion Transformers: Approximation, Estimation and Minimax Optimalitycited by 21
  11. 2024Scores as Actions: a framework of fine-tuning diffusion models by continuous-time reinforcement learningcited by 19
  12. 2025Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Accelerationcited by 17
  13. 2025Faster Diffusion Models via Higher-Order Approximationcited by 16
  14. 2024Regret of exploratory policy improvement and q-learningcited by 10
  15. 2026Conditional Diffusion Guidance under Hard Constraint: A Stochastic Analysis Approachcited by 10
  16. 2025Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Modelscited by 9
  17. 2024Structured Diffusion Models with Mixture of Gaussians as Prior Distributioncited by 8
  18. 2025Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Modelscited by 8
  19. 2025Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Boundscited by 8
  20. 2025Multimodal Atmospheric Super-Resolution With Deep Generative Modelscited by 8
  21. 2025Robust time series generation via Schr"odinger Bridge: a comprehensive evaluationcited by 7
  22. 2025Landing with the Score: Riemannian Optimization through Denoisingcited by 7
  23. 2025Understanding Sampler Stochasticity in Training Diffusion Models for RLHFcited by 7
  24. 2025Towards a unified framework for guided diffusion modelscited by 7
  25. 2025Malliavin Calculus for Score-based Diffusion Modelscited by 6
  26. 2026Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexitycited by 6
  27. 2024Diffusion Transformer Captures Spatial-Temporal Dependencies: A Theory for Gaussian Process Datacited by 5
  28. 2026Diffusion Models with Heavy-Tailed Targets: Score Estimation and Sampling Guaranteescited by 4
  29. 2025Diffusion Generative Models Meet Compressed Sensing, with Applications to Imaging and Financecited by 3
  30. 2026ART for Diffusion Sampling: A Reinforcement Learning Approach to Timestep Schedulecited by 3
  31. 2026Improved techniques for fine-tuning flow models via adjoint matching: a deterministic control pipelinecited by 3
  32. 2025Uncertainty quantification of neural network models of evolving processes via Langevin samplingcited by 2
  33. 2025Joint stochastic localization and applicationscited by 2
  34. 2025CINDES: Classification induced neural density estimator and simulatorcited by 1
  35. 2026Score Shocks: The Burgers Equation Structure of Diffusion Generative Modelscited by 1
  36. 2026A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Modelscited by 1
  37. 2025Diffusion Models under Alternative Noise: Simplified Analysis and Sensitivitycited by 0
  38. 2025Diffusion Transformers for Imputation: Statistical Efficiency and Uncertainty Quantificationcited by 0

50 most cited of 60 All 60 (JSON)

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  1. 2023Closing the ODE-SDE gap in score-based diffusion models through the Fokker-Planck equation
  2. 2025A PDE Perspective on Generative Diffusion Models
  3. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
  4. 2022Score-based Continuous-time Discrete Diffusion Models
  5. 2024Convergence of Score-Based Discrete Diffusion Models: A Discrete-Time Analysis
  6. 2022A Flexible Diffusion Model
  7. 2023Score-based Generative Modeling Through Backward Stochastic Differential Equations: Inversion and Generation
  8. 2022How Much is Enough? A Study on Diffusion Times in Score-based Generative Models