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.

通过将采样过程定义为扩展的逆时间随机微分方程 (ERSDE)，我们提出了 ER-SDE 求解器，为扩散模型 (DM) 的图像生成速度带来了突破性的提升，并在 ImageNet 64×64 数据集上实现了 3.45 FID 的效果。