The blooming diffusion probabilistic models (DPMs) have garnered significant interest due to their impressive performance and the elegant inspiration they draw from physics. While earlier DPMs relied upon the Markovian assumption, recent methods based on differential equations have been rapidly applied to enhance the efficiency and capabilities of these models. However, a theoretical interpretation encapsulating these diverse algorithms is insufficient yet pressingly required to guide further development of DPMs. In response to this need, we present FreeFlow, a framework that provides a thorough explanation of the diffusion formula as time-dependent optimal transport, where the evolutionary pattern of probability density is given by the gradient flows of a functional defined in Wasserstein space. Crucially, our framework necessitates a unified description that not only clarifies the subtle mechanism of DPMs but also indicates the roots of some defects through creative involvement of Lagrangian and Eulerian views to understand the evolution of probability flow. We particularly demonstrate that the core equation of FreeFlow condenses all stochastic and deterministic DPMs into a single case, showcasing the expansibility of our method. Furthermore, the Riemannian geometry employed in our work has the potential to bridge broader subjects in mathematics, which enable the involvement of more profound tools for the establishment of more outstanding and generalized models in the future.

FreeFlow是一个框架，以时间依赖的最优输运来解释扩散公式，其中概率密度的演化模式由Wasserstein空间中的泛函梯度流给出，既阐明了DPMs的微妙机制，也通过创造性地涉及Lagrangian和Eulerian观点来指示一些缺陷的根源，特别地，核心方程将所有随机和确定性DPMs凝聚为一个单一情况，展示了我们方法的可扩展性，此外，我们工作中使用的Riemannian几何有潜力桥接数学中更广泛的领域，为未来建立更优秀和更广义的模型提供了更深入的工具。

FreeFlow: 通过最优传输对扩散概率模型进行全面理解