This is a summary of unified framework of diffusion models by Stochastic Differential Equations (SDE) and the related topics.
In score-based diffusion models
The diffusion process ${\x_t}_{t=0}^T$ from the original input $\x_0$ to Gaussian noise $\x_T$ with continuous time variable $t\in\sbr{0,T}$ can be modeled by the Itô SDE
\[\begin{equation} d\x = \f(\x, t) dt + \bs{G}(\x, t)d\w_t \end{equation}\]We consider the case $\bs{G}(\x,t) = g(t)\I$ which is independent of $\x$. Then, Eq. (1) can be rewritten as
\[\begin{equation} d\x = f(t)\x dt + g(t)d\w_t \label{eq:forward-sde} \end{equation}\]The perturbation kernel of this SDE have the form
\[\begin{equation} p(\x_t \mid \x_0) = \N(\x_t \mid s_t\x_0, s_t^2\sigma_t^2\I) \end{equation}\]where
\[\begin{equation} s_t = \exp\nbr{\int_0^t f(\xi)d\xi} \quad \text{and}\quad \sigma_t^2 = \int_0^t\nbr{\frac{g(\xi)}{s(\xi)}}^2d\xi \end{equation}\]Here, I’ll use $s_t$ and $s(t)$, and $\sigma_t$ and $\sigma(t)$ interchangeably for simplicity. The corresponding marginal distribution is obtained by
\[\begin{align} p(\x_t) &= \int p(\x_t \mid \x_0)p_{data}(\x_0)d\x_0\\ &= s_t^{-d}\int p_{data}(\x_0)\N(\x_ts_t^{-1} \mid \x_0, \sigma_t^2\I)d\x_0\\ &= s_t^{-d}\int p_{data}(\x_0)\N(\x_ts_t^{-1}-\x_0 \mid 0, \sigma_t^2\I)d\x_0\\ &= s_t^{-d} \sbr{p_{data}(\x_0) * \N\nbr{0, \sigma_t^2\I}}(\x_ts_t^{-1})\\ &= s_t^{-d} p(\x_ts_t^{-1};\sigma_t) \label{eq:marginal} \end{align}\]The Fokker-Plank equation describes the evolution of the marginal distribution $p_t(\x)$, interchangeable with $p(\x_t)$, over time under the effect of drift forces and random (or noise) forces. It can be written as
\[\begin{align} \frac{\partial p_t(\x)}{\partial t} &= -\nabla_{\x}\sbr{\f(\x,t) p_t(\x)} + \frac{1}{2}\nabla_{\x}\nabla_{\x}\sbr{\bs{G}(\x,t)\bs{G}^T(\x,t)p_t(\x)}\\ &= -\sum_{i=1}^{d}\frac{\partial}{\partial x_i}\sbr{f_i(\x,t)p_t(\x)} + \frac{1}{2}\sum_{i=1}^{d}\sum_{j=1}^{d}\frac{\partial^2}{\partial \x_i \partial \x_j}\sbr{\sum_{k=1}^{d} G_{ik}(\x, t)G_{jk}(\x, t) p_t(\x)}\\ &= -\sum_{i=1}^{d}\frac{\partial}{\partial x_i}\sbr{f_i(\x,t)p_t(\x) -\frac{1}{2}\sbr{\nabla_{\x}\sbr{\bs{G}(\x,t)\bs{G}^T(\x,t)} + \bs{G}(\x,t)\bs{G}^T(\x,t)\nabla_{\x}\log p_t(\x)}p_t(\x)}\\ &= -\sum_{i=1}^{d}\frac{\partial}{\partial x_i}\sbr{\tilde{f}_i(\x,t)p_t(\x)} \label{eq:fokker-planck} \end{align}\]where
\[\tilde{f}_i(\x,t) = f_i(\x,t) - \frac{1}{2}\sbr{\nabla_{\x}\sbr{\bs{G}(\x,t)\bs{G}^T(\x,t)} + \bs{G}(\x,t)\bs{G}^T(\x,t)\nabla_{\x}\log p_t(\x)}.\]If we consider the case $\bs{G}(\x,t) = g(t)\I$ which is independent of $\x$, then Eq. $\ref{eq:fokker-planck}$ can be rewritten as
\[\begin{align} \frac{\partial p_t(\x)}{\partial t} &= -\nabla_{\x}\sbr{f(\x,t)p_t(\x)} + \frac{1}{2}g^2(t)\nabla_{\x}\nabla_{\x}\sbr{p_t(\x)}\\ &= -\sum_{i=1}^{d}\frac{\partial}{\partial x_i}\sbr{\sbr{f_i(\x,t)-\frac{1}{2}g^2(t)\nabla_{\x}\log p_t(\x)}p_t(\x)} \label{eq:fokker-planck-special} \end{align}\]According to the Fokker-Plank equation, there exists an ODE which shares the same marginal distribution $p(\x)$ as the SDE. From Eq. $\ref{eq:fokker-planck-special}$, the corresponding SDE is reduced to ODE given by
\[\begin{align} d\x &= \sbr{f(\x, t) - \frac{1}{2}g^2(t)\nabla_{\x}\log p_t(\x)}dt + 0d\w_t \\ &= \sbr{f(t)\x - \frac{1}{2}g^2(t)\nabla_{\x}\log p_t(\x)}dt \label{eq:prob-flow-ode} \end{align}\]The ODE is named as the probability flow ODE (PF ODE).
If we further build the ODE according to $s_t$ and $\sigma_t$, we have
\[\begin{equation} f(t) = \dot{s}_ts_t^{-1} \quad \text{and} \quad g_t = s_t\sqrt{2\dot{\sigma}_t\sigma_t}. \end{equation}\]The derivation can be found in EDM paper
where the marginal $p_t(\x) = s^{-1}_t p(\x_t s_t^{-1};\sigma_t)$ and
\[p(\x_ts_t^{-1}; \sigma_t) = \sbr{p_{data}(\x_0) * \N\nbr{0, \sigma_t^2\I}}(\x_ts_t^{-1})\]as shown in Eq. $\ref{eq:marginal}$.
According to the Eq. (102) in
The PF ODE is a special case of the SDE when $g(t) = 0$. The derivation is given in the EDM paper
where $\beta(t)$ is a free function.
The reverse diffusion process from $\x_T$ to $\x_0$ can also be modeled by Itô SDE according to