In applications of diffusion models, controllable generation is of practical significance, but is also challenging. Current methods for controllable generation primarily focus on modifying the score function of diffusion models, while Mean Reverting (MR) Diffusion directly modifies the structure of the stochastic differential equation (SDE), making the incorporation of image conditions simpler and more natural. However, current training-free fast samplers are not directly applicable to MR Diffusion. And thus MR Diffusion requires hundreds of NFEs (number of function evaluations) to obtain high-quality samples. In this paper, we propose a new algorithm named MaRS (MR Sampler) to reduce the sampling NFEs of MR Diffusion. We solve the reverse-time SDE and the probability flow ordinary differential equation (PF-ODE) associated with MR Diffusion, and derive semi-analytical solutions. The solutions consist of an analytical function and an integral parameterized by a neural network. Based on this solution, we can generate high-quality samples in fewer steps. Our approach does not require training and supports all mainstream parameterizations, including noise prediction, data prediction and velocity prediction. Extensive experiments demonstrate that MR Sampler maintains high sampling quality with a speedup of 10 to 20 times across ten different image restoration tasks. Our algorithm accelerates the sampling procedure of MR Diffusion, making it more practical in controllable generation.

翻译：在扩散模型的应用中，可控生成具有重要的实际意义，但也极具挑战性。当前的可控生成方法主要集中于修改扩散模型的分数函数，而均值回复扩散则直接修改随机微分方程的结构，使得图像条件的融入更为简单自然。然而，当前无需训练的快采样器无法直接应用于均值回复扩散，因此均值回复扩散需要数百次函数评估才能获得高质量样本。本文提出一种名为MaRS的新算法，旨在减少均值回复扩散的采样函数评估次数。我们求解了与均值回复扩散相关的反向时间随机微分方程和概率流常微分方程，并推导出半解析解。该解由一个解析函数和一个由神经网络参数化的积分项组成。基于此解，我们可以在更少的步数内生成高质量样本。我们的方法无需训练，并支持所有主流参数化形式，包括噪声预测、数据预测和速度预测。大量实验表明，MR采样器在十种不同的图像恢复任务中，能以10至20倍的加速比保持高采样质量。我们的算法加速了均值回复扩散的采样过程，使其在可控生成中更具实用性。