# On the Trajectory Regularity of ODE-based Diffusion Sampling

arXiv: https://arxiv.org/abs/2405.11326 · HTML: https://mlsanity.com/paper/2405.11326/

## 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.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 49)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [DreamBooth: Fine Tuning Text-to-Image Diffusion Models for Subject-Driven Generation](https://mlsanity.com/paper/2208.12242/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2304.08818/) (2023)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [eDiff-I: Text-to-Image Diffusion Models with an Ensemble of Expert Denoisers](https://mlsanity.com/paper/2211.01324/) (2022)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)

## Built on by (most cited first, 30 of 55)

- [Learning to Discretize Denoising Diffusion ODEs](https://mlsanity.com/paper/2405.15506/) (2024)
- [Simple and Fast Distillation of Diffusion Models](https://mlsanity.com/paper/2409.19681/) (2024)
- [Schedule On the Fly: Diffusion Time Prediction for Faster and Better Image Generation](https://mlsanity.com/paper/2412.01243/) (2024)
- [Conditional Image Synthesis with Diffusion Models: A Survey](https://mlsanity.com/paper/2409.19365/) (2024)
- [OmniCache: A Trajectory-Oriented Global Perspective on Training-Free Cache Reuse for Diffusion Transformer Models](https://mlsanity.com/paper/2508.16212/) (2025)
- [Lipschitz-Guided Design of Interpolation Schedules in Generative Models](https://mlsanity.com/paper/2509.01629/) (2025)
- [Spectral Analysis of Diffusion Models with Application to Schedule Design](https://mlsanity.com/paper/2502.00180/) (2025)
- [Distilling Parallel Gradients for Fast ODE Solvers of Diffusion Models](https://mlsanity.com/paper/2507.14797/) (2025)
- [S4S: Solving for a Diffusion Model Solver](https://mlsanity.com/paper/2502.17423/) (2025)
- [Learning to Integrate Diffusion ODEs by Averaging the Derivatives](https://mlsanity.com/paper/2505.14502/) (2025)
- [EVODiff: Entropy-aware Variance Optimized Diffusion Inference](https://mlsanity.com/paper/2509.26096/) (2025)
- [PFDiff: Training-Free Acceleration of Diffusion Models Combining Past and Future Scores](https://mlsanity.com/paper/2408.08822/) (2024)
- [Denoising as Path Planning: Training-Free Acceleration of Diffusion Models with DPCache](https://mlsanity.com/paper/2602.22654/) (2026)
- [Optimal Stepsize for Diffusion Sampling](https://mlsanity.com/paper/2503.21774/) (2025)
- [B'ezierFlow: Learning B'ezier Stochastic Interpolant Schedulers for Few-Step Generation](https://mlsanity.com/paper/2512.13255/) (2025)
- [Differentiable Solver Search for Fast Diffusion Sampling](https://mlsanity.com/paper/2505.21114/) (2025)
- [Adaptive Stochastic Coefficients for Accelerating Diffusion Sampling](https://mlsanity.com/paper/2510.23285/) (2025)
- [Unleashing High-Quality Image Generation in Diffusion Sampling Using Second-Order Levenberg-Marquardt-Langevin](https://mlsanity.com/paper/2505.24222/) (2025)
- [DiFiC: Your Diffusion Model Holds the Secret to Fine-Grained Clustering](https://mlsanity.com/paper/2412.18838/) (2024)
- [Improving Diffusion-based Inverse Algorithms under Few-Step Constraint via Learnable Linear Extrapolation](https://mlsanity.com/paper/2503.10103/) (2025)
- [Geometric Regularity in Deterministic Sampling Dynamics of Diffusion-based Generative Models](https://mlsanity.com/paper/2506.10177/) (2025)
- [DyWeight: Dynamic Gradient Weighting for Few-Step Diffusion Sampling](https://mlsanity.com/paper/2603.11607/) (2026)
- [Few-Step Diffusion Sampling Through Instance-Aware Discretizations](https://mlsanity.com/paper/2603.17671/) (2026)
- [Diffusion Sampling Correction via Approximately 10 Parameters](https://mlsanity.com/paper/2411.06503/) (2024)
- [Image Diffusion Preview with Consistency Solver](https://mlsanity.com/paper/2512.13592/) (2025)
- [Parallel Diffusion Solver via Residual Dirichlet Policy Optimization](https://mlsanity.com/paper/2512.22796/) (2025)
- [Mitigating the Contractivity Trap in Diffusion ODEs via Stein Stabilization](https://mlsanity.com/paper/2606.07835/) (2026)
- [Distilling Drifting Transformers with Representation Autoencoders](https://mlsanity.com/paper/2606.15553/) (2026)
- [DICE: Distilling Classifier-Free Guidance into Text Embeddings](https://mlsanity.com/paper/2502.03726/) (2025)
- [On the Separability of Information in Diffusion Models](https://mlsanity.com/paper/2509.23937/) (2025)

## 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.