Drawing from the theory of stochastic differential equations, we introduce a novel sampling method for known distributions and a new algorithm for diffusion generative models with unknown distributions. Our approach is inspired by the concept of the reverse diffusion process, widely adopted in diffusion generative models. Additionally, we derive the explicit convergence rate based on the smooth ODE flow. For diffusion generative models and sampling, we establish a {\it dimension-free} particle approximation convergence result. Numerical experiments demonstrate the effectiveness of our method. Notably, unlike the traditional Langevin method, our sampling method does not require any regularity assumptions about the density function of the target distribution. Furthermore, we also apply our method to optimization problems.

翻译：基于随机微分方程理论，我们针对已知分布提出了一种新颖的采样方法，并为未知分布的扩散生成模型设计了一种新算法。该方法受到扩散生成模型中广泛采用的反向扩散过程概念的启发。此外，我们基于光滑常微分方程流推导出了显式收敛速率。对于扩散生成模型与采样问题，我们建立了{\it 维度无关}的粒子逼近收敛结果。数值实验验证了本方法的有效性。值得注意的是，与传统朗之万方法不同，我们的采样方法无需对目标分布密度函数施加任何正则性假设。此外，我们还将该方法应用于优化问题。