Gradient flow
Prelimiaries
Flux: The amount of mass ($q$) pass through a unit area per unit time ($kg/m^2/s$). Or informally, it’s a product of density of the fluid times a n-D vector of velocity on a topological surface:
\[\vec{j} = \rho \vec{u}\]$\text{where }\vec{u}\text{ is the velocity field.}$
Continuity equation: the transport process of mass of conserved material.
\[\frac{\partial{\rho}}{\partial{t}} + \nabla \vec{j} = \sigma ,\] \[\text{where density, divergence, flux and generation of mass is noted as } \rho, \nabla, \vec{j}, \text{ and } \sigma.\]Probability perspective
As an analogy to mass conservation in physics world, probability mass is also considered conservative when being mapped from one to another.
Evolution of pdf. Here, the overall probability mass of a moving distribution ($\mathbb{Q}$) is mapped to the target distribution ($\mathbb{P}$). The probability density, $q_{t}(x)$, is analogous to the density of mass which evolves overtime (${q_{t}}\text{, }t\in \mathbb{R}^{+}$), and finally mapped to anothre probability density ($p$).
Metrics space. Unlike the cartisian coordinates in physics space, the “flux” in the measurement context refers to:
\[\begin{split}-\text{flux}&=-\nabla f(x_{t})\\&=\text{div}(q_{t}\nabla_{metircs}\mathcal{F}(q_{t}))\\&=\text{div}(q_{t}\nabla_{x}\frac{\partial{\mathcal{F(q_{t})}}}{\partial{q_{t}}}) \end{split}\]$\text{where the last equation can be seen as the gradient of the measurement }\mathcal{F(\cdot)}\text{ on Wasserstein space}$