  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.
