mlsanityData to 14 Oct 2026

On the Trajectory Regularity of ODE-based Diffusion Sampling

arXiv 2405.11326 ↗First posted 18 May 2024cs.LG cs.CVcited by 55

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion-based generative models use stochastic differential equations (SDEs) and their equivalent ordinary differential equations (ODEs) to establish a smooth connection between a complex data distribution and a tractable prior distribution. In this paper, we identify several intriguing trajectory properties in the ODE-based sampling process of diffusion models. We characterize an implicit denoising trajectory and discuss its vital role in forming the coupled sampling trajectory with a strong shape regularity, regardless of the generated content. We also describe a dynamic programming-based scheme to make the time schedule in sampling better fit the underlying trajectory structure. This simple strategy requires minimal modification to any given ODE-based numerical solvers and incurs negligible computational cost, while delivering superior performance in image generation, especially in 5~ 10 function evaluations.

Key facts. Cited by 55 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 49 of its 78 references that are in the map. First posted on arXiv 18 May 2024.

Builds on

49 of 78 references.

Closest ancestors · 34

Same direction, newest first

  1. 2024Align Your Steps: Optimizing Sampling Schedules in Diffusion Models22 Apr
  2. 2024Scaling Rectified Flow Transformers for High-Resolution Image Synthesis5 Mar
  3. 2023Fast ODE-based Sampling for Diffusion Models in Around 5 Steps30 Nov
  4. 2023Improved Order Analysis and Design of Exponential Integrator for Diffusion Models Sampling4 Aug
  5. 2023SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis4 Jul
  6. 2023A Geometric Perspective on Diffusion Models31 May
  7. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers6 Mar
  8. 2023Consistency Models2 Mar
  9. 2023UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models9 Feb
  10. 2023Stable Target Field for Reduced Variance Score Estimation in Diffusion Models1 Feb
  11. 2022Fast Sampling of Diffusion Models via Operator Learning24 Nov
  12. 2022DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models2 Nov
22 more
  1. 2022Diffusion Models already have a Semantic Latent Space20 Oct
  2. 2022GENIE: Higher-Order Denoising Diffusion Solvers11 Oct
  3. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  4. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  5. 2022Score-Based Generative Models Detect Manifolds2 Jun
  6. 2022DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps2 Jun
  7. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  8. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  9. 2022Pseudo Numerical Methods for Diffusion Models on Manifolds20 Feb
  10. 2022Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models17 Jan
  11. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  12. 2021Variational Diffusion Models1 Jul
  13. 2021Score-based Generative Modeling in Latent Space10 Jun
  14. 2021Learning to Efficiently Sample from Diffusion Probabilistic Models7 Jun
  15. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  16. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  17. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  18. 2021Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed7 Jan
  19. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  20. 2020Denoising Diffusion Implicit Models6 Oct
  21. 2020Denoising Diffusion Probabilistic Models19 Jun
  22. 2020Improved Techniques for Training Score-Based Generative Models16 Jun

Other references · 15

  1. 2014Microsoft COCO: Common Objects in Contextcited by 17,608
  2. 2014ImageNet Large Scale Visual Recognition Challengecited by 8,498
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  4. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,585
  5. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,355
  6. 2018A Style-Based Generator Architecture for Generative Adversarial Networkscited by 6,013
  7. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  8. 2022DreamBooth: Fine Tuning Text-to-Image Diffusion Models for Subject-Driven Generationcited by 3,300
  9. 2022Video Diffusion Modelscited by 2,305
  10. 2023Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Modelscited by 1,476
  11. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,354
  12. 2022eDiff-I: Text-to-Image Diffusion Models with an Ensemble of Expert Denoiserscited by 931
3 more
  1. 2022PhysDiff: Physics-Guided Human Motion Diffusion Modelcited by 368
  2. 2013Generalized Denoising Auto-Encoders as Generative Modelscited by 113
  3. 2015A review of mean-shift algorithms for clusteringcited by 19

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 8; 2025: 20; 2026: 27.

55 papers.

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  1. 2024Learning to Discretize Denoising Diffusion ODEscited by 41
  2. 2024Simple and Fast Distillation of Diffusion Modelscited by 32
  3. 2024Schedule On the Fly: Diffusion Time Prediction for Faster and Better Image Generationcited by 30
  4. 2024Conditional Image Synthesis with Diffusion Models: A Surveycited by 29
  5. 2025OmniCache: A Trajectory-Oriented Global Perspective on Training-Free Cache Reuse for Diffusion Transformer Modelscited by 18
  6. 2025Lipschitz-Guided Design of Interpolation Schedules in Generative Modelscited by 17
  7. 2025Spectral Analysis of Diffusion Models with Application to Schedule Designcited by 13
  8. 2025Distilling Parallel Gradients for Fast ODE Solvers of Diffusion Modelscited by 13
  9. 2025S4S: Solving for a Diffusion Model Solvercited by 10
  10. 2025Learning to Integrate Diffusion ODEs by Averaging the Derivativescited by 10
  11. 2025EVODiff: Entropy-aware Variance Optimized Diffusion Inferencecited by 10
  12. 2024PFDiff: Training-Free Acceleration of Diffusion Models Combining Past and Future Scorescited by 8
38 more
  1. 2026Denoising as Path Planning: Training-Free Acceleration of Diffusion Models with DPCachecited by 8
  2. 2025Optimal Stepsize for Diffusion Samplingcited by 7
  3. 2025B'ezierFlow: Learning B'ezier Stochastic Interpolant Schedulers for Few-Step Generationcited by 7
  4. 2025Differentiable Solver Search for Fast Diffusion Samplingcited by 6
  5. 2025Adaptive Stochastic Coefficients for Accelerating Diffusion Samplingcited by 6
  6. 2025Unleashing High-Quality Image Generation in Diffusion Sampling Using Second-Order Levenberg-Marquardt-Langevincited by 5
  7. 2024DiFiC: Your Diffusion Model Holds the Secret to Fine-Grained Clusteringcited by 4
  8. 2025Improving Diffusion-based Inverse Algorithms under Few-Step Constraint via Learnable Linear Extrapolationcited by 4
  9. 2025Geometric Regularity in Deterministic Sampling Dynamics of Diffusion-based Generative Modelscited by 4
  10. 2026DyWeight: Dynamic Gradient Weighting for Few-Step Diffusion Samplingcited by 4
  11. 2026Few-Step Diffusion Sampling Through Instance-Aware Discretizationscited by 4
  12. 2024Diffusion Sampling Correction via Approximately 10 Parameterscited by 3
  13. 2025Image Diffusion Preview with Consistency Solvercited by 3
  14. 2025Parallel Diffusion Solver via Residual Dirichlet Policy Optimizationcited by 3
  15. 2026Mitigating the Contractivity Trap in Diffusion ODEs via Stein Stabilizationcited by 3
  16. 2026Distilling Drifting Transformers with Representation Autoencoderscited by 3
  17. 2025DICE: Distilling Classifier-Free Guidance into Text Embeddingscited by 2
  18. 2025On the Separability of Information in Diffusion Modelscited by 2
  19. 2026Sharpen Your Flow: Sharpness-Aware Sampling for Flow Matchingcited by 2
  20. 2024On the Relation Between Linear Diffusion and Power Iterationcited by 1
  21. 2026Analyzing and Guiding Zero-Shot Posterior Sampling in Diffusion Modelscited by 1
  22. 2026Analyzing and Improving Fast Sampling of Text-to-Image Diffusion Modelscited by 1
  23. 2026Improving Diffusion Generalization with Weak-to-Strong Segmented Guidancecited by 1
  24. 2026Three Creates All: You Only Sample 3 Stepscited by 1
  25. 2025A Geometric Unification of Generative AI with Manifold-Probabilistic Projection Modelscited by 0
  26. 2025GAS: Improving Discretization of Diffusion ODEs via Generalized Adversarial Solvercited by 0
  27. 2026A Gaussian Perspective for Distributional Discrepancy in Generative Diffusion Modelscited by 0
  28. 2026Information-geometric adaptive sampling for graph diffusioncited by 0
  29. 2026Tracing the Oracle: Improving Diffusion Timestep Scheduling for 3D CT Reconstructioncited by 0
  30. 2026Learning to Solve Generative ODEs Beyond the Linear Spancited by 0
  31. 2026Conditioning Matters: Stabilizing Inversion and Attention in Diffusion Image Editingcited by 0
  32. 2026D2PO: Optimizing Diffusion Samplers via Dynamic Preferencecited by 0
  33. 2026Improved denoising diffusion probabilistic models with efficient non-diagonal covariance modelingcited by 0
  34. 2026Interpreting Object-Dependent Concept Brittleness in Text-to-Image Diffusion Modelscited by 0
  35. 2026Model-Aware Schedules Improve Generation via Fiberwise Optimal Transportcited by 0
  36. 2026RAIN: Region-Aware Inversion Network for Semantic Watermark Extractioncited by 0
  37. 2026I-SplineFlow: Learning Monotone Spline Stochastic Interpolant Schedulers for Few-Step Generationcited by 0
  38. 2026Distill Locally, Schedule Globally: Flow Maps for Few-Step Text-to-Speechcited by 0

50 most cited of 55 All 55 (JSON)

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  1. 2024Input-Adaptive Generative Dynamics in Diffusion Models
  2. 2024New algorithms for sampling and diffusion models
  3. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
  4. 2023Exploring the Optimal Choice for Generative Processes in Diffusion Models: Ordinary vs Stochastic Differential Equations
  5. 2022An optimal control perspective on diffusion-based generative modeling
  6. 2024Diffusion Bridge Implicit Models
  7. 2026Formalizing the Sampling Design Space of Diffusion-Based Generative Models via Adaptive Solvers and Wasserstein-Bounded Timesteps
  8. 2025Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Models