In our recent studies on the diffusion-based generative models, we need to solve an inverse-time process of the diffusion model. Specifically, the inverse-time process of interest can be formulated as the following SDE:
$$\text{d}\vec{\mathbf{X}}_t= -e^{-t}\cdot\text{tanh}\left(e^{-t}\left<\vec{\mathbf{X}}_t,\vec{{\mathbf{\Theta}}}\right>\right)\cdot\vec{\mathbf{\Theta}}\text{d}t+\sqrt{2}\text{d}\vec{\mathbf{W}}_t,~~~~~~~~~~~(1)$$
with initialization at time $T>0$ given as $\vec{\mathbf{X}}_{T}=\vec{\mathbf{x}}_{T}\in\mathbb{R}^{d}$, where $\vec{{\mathbf{\Theta}}}\in\mathbb{R}^{d}$ is a given vector, and $\vec{\mathbf{W}}_t$ is a standard inver-time $d$-dimensional Wiener process.
Out goal is to get an analytic expression of $\vec{\mathbf{X}}_{0}$. However, we do not know much about how to solve the SDE in Eq. (1).
We instead tried to consider the corresponding Fokker-Planck equation of Eq. (1) as follows:
\begin{align} \frac{\partial p_t(\vec{\mathbf{X}})}{\partial t} =&e^{-2t}\|\vec{\mathbf{\Theta}}\|_2^2\cdot\text{sech}^2\left(e^{-t}\left<\vec{\mathbf{X}},\vec{\mathbf{\Theta}}\right>\right)p_t(\vec{\mathbf{X}})\\ &+e^{-t}\text{tanh}\left(e^{-t}\left<\vec{\mathbf{X}}_t,\vec{{\mathbf{\Theta}}}\right>\right)\left<\vec{{\mathbf{\Theta}}},\nabla_{\vec{\mathbf{X}}}p_t(\vec{\mathbf{X}})\right>+\sum_{i,j}\frac{\partial^2}{\partial x_i \partial x_j}p_t(\vec{\mathbf{X}}).~~~~~~~(2) \end{align}
However, this is a seemingly complicated second-order partial differential equation, and we have very little experience in solving it.
Could you please give us some hints or references that can help us solve Eqs. (1) or (2)?