Diffusion models (DMs) demonstrate potent image generation capabilities in various generative modeling tasks. Nevertheless, their primary limitation lies in slow sampling speed, requiring hundreds or thousands of sequential function evaluations through large neural networks to generate high-quality images. Sampling from DMs can be seen as solving corresponding stochastic differential equations (SDEs) or ordinary differential equations (ODEs). In this work, we formulate the sampling process as an extended reverse-time SDE (ER SDE), unifying prior explorations into ODEs and SDEs. Leveraging the semi-linear structure of ER SDE solutions, we offer exact solutions and arbitrarily high-order approximate solutions for VP SDE and VE SDE, respectively. Based on the solution space of the ER SDE, we yield mathematical insights elucidating the superior performance of ODE solvers over SDE solvers in terms of fast sampling. Additionally, we unveil that VP SDE solvers stand on par with their VE SDE counterparts. Finally, we devise fast and training-free samplers, ER-SDE Solvers, elevating the efficiency of stochastic samplers to unprecedented levels. Experimental results demonstrate achieving 3.45 FID in 20 function evaluations and 2.24 FID in 50 function evaluations on the ImageNet 64$\times$64 dataset.

翻译：扩散模型（DMs）在各类生成建模任务中展现出强大的图像生成能力，但其主要局限在于采样速度缓慢：生成高质量图像需通过大型神经网络执行数百至数千次顺序函数评估。从DMs采样可视为求解相应随机微分方程（SDEs）或常微分方程（ODEs）。本研究将采样过程形式化为扩展反时序SDE（ER SDE），统一了以往对ODEs和SDEs的探索。利用ER SDE解的半线性结构，我们分别针对VP SDE和VE SDE给出了精确解及任意高阶近似解。基于ER SDE的解空间，我们获得数学洞见，阐明了ODE求解器在快速采样中优于SDE求解器的根本原因。此外，我们发现VP SDE求解器的表现与VE SDE求解器相当。最终，我们设计了快速且无需训练的采样器ER-SDE求解器，将随机采样器的效率提升至空前水平。实验证明，在ImageNet 64×64数据集上，该方法在20次函数评估下达到3.45 FID，50次函数评估下达到2.24 FID。