Elucidating the Solution Space of Extended Reverse-Time SDE for Diffusion Models

Qinpeng Cui, Xinyi Zhang, Qiqi Bao, Qingmin Liao; Proceedings of the Winter Conference on Applications of Computer Vision (WACV), 2025, pp. 243-252

Sampling from Diffusion Models can alternatively be seen as solving differential equations where there is a challenge in balancing speed and image visual quality. ODE-based samplers offer rapid sampling time but reach a performance limit whereas SDE-based samplers achieve superior quality albeit with longer iterations. In this work we formulate the sampling process as an Extended Reverse-Time SDE (ER SDE) unifying prior explorations into ODEs and SDEs. Theoretically leveraging the semi-linear structure of ER SDE solutions we offer exact solutions and approximate solutions for VP SDE and VE SDE respectively. Based on the approximate solution space of the ER SDE referred to as one-step prediction errors we yield mathematical insights elucidating the rapid sampling capability of ODE solvers and the high-quality sampling ability of SDE solvers. Additionally we unveil that VP SDE solvers stand on par with their VE SDE counterparts. Based on these findings leveraging the dual advantages of ODE solvers and SDE solvers we devise efficient high-quality samplers namely ER-SDE-Solvers. Experimental results demonstrate that ER-SDE-Solvers achieve state-of-the-art performance across all stochastic samplers while maintaining efficiency of deterministic samplers. Specifically on the ImageNet 128 x 128 dataset ER-SDE-Solvers obtain 8.33 FID in only 20 function evaluations. Code is available at https://github.com/QinpengCui/ER-SDE-Solver