# SEEDS: Exponential SDE Solvers for Fast High-Quality Sampling from Diffusion Models arXiv: https://arxiv.org/abs/2305.14267 · HTML: https://mlsanity.com/paper/2305.14267/ ## Key facts - Cited by 36 papers in the map (data through 14 Oct 2026). - Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. - Builds on 19 of its 26 references that are in the map. - First posted on arXiv 23 May 2023. ## Placed in - Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/) ## Builds on (most cited first, 19 of 19) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020) - [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) - [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015) - [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021) - [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022) - [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022) - [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022) - [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021) - [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020) - [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022) - [Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speech](https://mlsanity.com/paper/2105.06337/) (2021) - [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022) - [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021) - [Gotta Go Fast When Generating Data with Score-Based Models](https://mlsanity.com/paper/2105.14080/) (2021) - [GENIE: Higher-Order Denoising Diffusion Solvers](https://mlsanity.com/paper/2210.05475/) (2022) - [gDDIM: Generalized denoising diffusion implicit models](https://mlsanity.com/paper/2206.05564/) (2022) - [WaveGrad 2: Iterative Refinement for Text-to-Speech Synthesis](https://mlsanity.com/paper/2106.09660/) (2021) ## Built on by (most cited first, 30 of 36) - [DiffusionNFT: Online Diffusion Reinforcement with Forward Process](https://mlsanity.com/paper/2509.16117/) (2025) - [Masked Diffusion Models are Secretly Time-Agnostic Masked Models and Exploit Inaccurate Categorical Sampling](https://mlsanity.com/paper/2409.02908/) (2024) - [All-in-one simulation-based inference](https://mlsanity.com/paper/2404.09636/) (2024) - [Boosting Diffusion Models with Moving Average Sampling in Frequency Domain](https://mlsanity.com/paper/2403.17870/) (2024) - [Diffusion Bridge Implicit Models](https://mlsanity.com/paper/2405.15885/) (2024) - [Fast Solvers for Discrete Diffusion Models: Theory and Applications of High-Order Algorithms](https://mlsanity.com/paper/2502.00234/) (2025) - [Stochastic Runge-Kutta Methods: Provable Acceleration of Diffusion Models](https://mlsanity.com/paper/2410.04760/) (2024) - [Simple and Fast Distillation of Diffusion Models](https://mlsanity.com/paper/2409.19681/) (2024) - [Schrodinger Bridges Beat Diffusion Models on Text-to-Speech Synthesis](https://mlsanity.com/paper/2312.03491/) (2023) - [Consistency Diffusion Bridge Models](https://mlsanity.com/paper/2410.22637/) (2024) - [Gaussian Mixture Solvers for Diffusion Models](https://mlsanity.com/paper/2311.00941/) (2023) - [Fast Samplers for Inverse Problems in Iterative Refinement Models](https://mlsanity.com/paper/2405.17673/) (2024) - [SCott: Accelerating Diffusion Models with Stochastic Consistency Distillation](https://mlsanity.com/paper/2403.01505/) (2024) - [Taming Diffusion Prior for Image Super-Resolution with Domain Shift SDEs](https://mlsanity.com/paper/2409.17778/) (2024) - [Efficient Integrators for Diffusion Generative Models](https://mlsanity.com/paper/2310.07894/) (2023) - [EVODiff: Entropy-aware Variance Optimized Diffusion Inference](https://mlsanity.com/paper/2509.26096/) (2025) - [Elucidating the solution space of extended reverse-time SDE for diffusion models](https://mlsanity.com/paper/2309.06169/) (2023) - [AdjointDEIS: Efficient Gradients for Diffusion Models](https://mlsanity.com/paper/2405.15020/) (2024) - [Sublinear iterations can suffice even for DDPMs](https://mlsanity.com/paper/2511.04844/) (2025) - [Strong Stochastic Flow Maps](https://mlsanity.com/paper/2606.01086/) (2026) - [SHIFT: Stochastic Hidden-Trajectory Deflection for Removing Diffusion-based Watermark](https://mlsanity.com/paper/2603.29742/) (2026) - [Hierarchical Schedule Optimization for Fast and Robust Diffusion Model Sampling](https://mlsanity.com/paper/2511.11688/) (2025) - [Mitigating the Contractivity Trap in Diffusion ODEs via Stein Stabilization](https://mlsanity.com/paper/2606.07835/) (2026) - [Greed is Good: A Unifying Perspective on Guided Generation](https://mlsanity.com/paper/2502.08006/) (2025) - [Flow Diverse and Efficient: Learning Momentum Flow Matching via Stochastic Velocity Field Sampling](https://mlsanity.com/paper/2506.08796/) (2025) - [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025) - [Learning Energy-Based Models from Stochastic Interpolants using Spatiotemporal Differences](https://mlsanity.com/paper/2605.26850/) (2026) - [Multi-GradSpeech: Towards Diffusion-based Multi-Speaker Text-to-speech Using Consistent Diffusion Models](https://mlsanity.com/paper/2308.10428/) (2023) - [Elucidating the Preconditioning in Consistency Distillation](https://mlsanity.com/paper/2502.02922/) (2025) - [DiFA: Inference-Time Forward-Process Alignment for Diffusion Models](https://mlsanity.com/paper/2607.17972/) (2026) ## Abstract (arXiv) A potent class of generative models known as Diffusion Probabilistic Models (DPMs) has become prominent. A forward diffusion process adds gradually noise to data, while a model learns to gradually denoise. Sampling from pre-trained DPMs is obtained by solving differential equations (DE) defined by the learnt model, a process which has shown to be prohibitively slow. Numerous efforts on speeding-up this process have consisted on crafting powerful ODE solvers. Despite being quick, such solvers do not usually reach the optimal quality achieved by available slow SDE solvers. Our goal is to propose SDE solvers that reach optimal quality without requiring several hundreds or thousands of NFEs to achieve that goal. We propose Stochastic Explicit Exponential Derivative-free Solvers (SEEDS), improving and generalizing Exponential Integrator approaches to the stochastic case on several frameworks. After carefully analyzing the formulation of exact solutions of diffusion SDEs, we craft SEEDS to analytically compute the linear part of such solutions. Inspired by the Exponential Time-Differencing method, SEEDS use a novel treatment of the stochastic components of solutions, enabling the analytical computation of their variance, and contains high-order terms allowing to reach optimal quality sampling ~3-5× faster than previous SDE methods. We validate our approach on several image generation benchmarks, showing that SEEDS outperform or are competitive with previous SDE solvers. Contrary to the latter, SEEDS are derivative and training free, and we fully prove strong convergence guarantees for them.