In the previous posting (link), I introuced forward stochastic differential equation (SDE) that transforms a data sample $x_0$ into Gaussian noise $x_T$
dx_t = f(x_t, t) dt + g(t) dw_t, \quad 0\leq t \leq T \tag{Forward SDE}where x_t, f(x_t,t) \in \mathbb{R}^d , g(t) \in \mathbb{R} and d denotes the data dimension.
In a diffusion model, data samples are generated by reverse SDE starting from a state x_T drawn from a simple distribution (for example \mathcal{N}(0,\mathbf{I}) at t=T
dx_t = \left[f(x_t, t) - g(t)^2 \nabla_{x_t} \log p_t (x_t) \right]dt + g(t) d\bar{w}_t, \tag{Reverse SDE}where \bar{w}_t is a reverse Brownian motion and \log p_t (x_t) , called the score function, is the logarithm of probability density of x_t .