VP-SDE that we have

$dx=-\frac12\beta_txdt+\sqrt{\beta_t}dw$

PF-ODE

from the sde

$dx=f(x,t)dt+g(t)dWt$

by Fokker-planck

$\frac{\partial p_t(x)}{\partial t}=-\frac{\partial}{\partial x}[f(x,t)p_t(x)]+\frac12g(t)^2\frac{\partial^2p_t(x)}{\partial x^2}$

Set the Probability Flow ODE as:

$dx=u(x,t)dt$ “Where diffusion term is 0”

Then by fokker-planck equation.

$\frac{\partial p_t(x)}{\partial t}=-\frac{\partial}{\partial x}[u(x,t)p_t(x)]$

The probability flow of SDE and ODE have to same, then:

$-\frac{\partial}{\partial x}[u(x,t)p_t(x)]=-\frac{\partial}{\partial x}[f(x,t)p_t(x)]+\frac12g(t)^2\frac{\partial^2p_t(x)}{\partial x^2}$

log differentiate property: $\frac{\partial\log p_t(x)}{\partial x}=\frac{1}{p_t(x)}\frac{\partial p_t(x)}{\partial x}:p_t(x)\frac{\partial\log p_t(x)}{\partial x}=\frac{}{}\frac{\partial p_t(x)}{\partial x}$

$\frac12g(t)^2\frac{\partial^2p_t(x)}{\partial x^2}=\frac{\partial}{\partial x}[\frac12g(t)^2p_t(x)\frac{\partial \log p_t(x)}{\partial x}]$