\section{System Configurations}\label{appendix:system}
Here, we report system configurations of all the conducted numerical experiments.

\subsection{1D Linear}\label{appendix:1D-L-sys}
We consider an 1D linear system (Ornstein-Uhlenbech process) in \citet[Eq. 4.19]{pavliotis2014stochastic}
$$dx = -\beta x dt + \sqrt{D} dw.$$
From the analytical solution in \citet[Eq. 4.22]{pavliotis2014stochastic}:
$$p(x,t|x_0)=\sqrt{\frac{\beta}{2 \pi D(1-e^{-2 \beta t})}} \exp \Big(-\frac{\beta(x-x_0e^{-\beta t})^2}{2D(1-e^{-2\beta t})} \Big),$$ 
we set the initial distribution as $p_0(x) = p(x,t=1|x_0=1)$, and the system parameters are $\beta=D=0.2$. The computation domain is $t\in[1,3]$ and $x\in[-6,6]$.

\subsection{1D Nonlinear}\label{appendix:1D-NL-sys}
We consider an 1D nonlinear system
$$ dx = (ax^3+bx^2+cx+d)dt + edw.$$
The initial distribution is Gaussian
$$p_0(x)\sim \mathcal{N}(\mu,\sigma^2).$$
The computation domain is $t \in [0,5]$ and $x\in[-6,6]$, and the system parameters are $a=-0.1,b=0.1,c=0.5,d=0.5,e=0.8,\mu=-2$ and $\sigma=0.5$.
Since there is no analytical solution, the "true" PDF $p(x,t)$ is obtained by extensive Monte-Carlo simulation using Euler Scheme with small integration time step $\Delta t = 0.0005$ and $10^8$ samples.

\subsection{2D Inverted Pendulum}\label{appendix:2D-PEND-sys}
We consider 2D nonlinear system of inverted pendulum
\begin{align*}
    d x_1 &= x_2 dt + B_{11} dw, \\
    d x_2 &= -\frac{g}{l}\sin(x_1) dt + B_{22} dw,
\end{align*}
where $x_1$ is the angle, and $x_2$ is the angular rate. The initial distribution is multivariate Gaussian
$$
p_0(x)\sim \mathcal{N}(\mu, \Sigma).
$$
The computation domain is $t \in [0,5]$ and $x_1,x_2 \in [-3 \pi, 3\pi]$, and the system parameters are $g=9.8,l=9.8,B_{11}=B_{22}=0.5,\mu=[0.5\pi,0]^T$ and $diag([0.5,0.5]$.
Similarly, the "true" PDF $p(x,t)$ is obtained by Monte-Carlo simulation using Euler Scheme with integration time step $\Delta t=0.01$ and $2 \times 10^7$ samples.

\subsection{2D Duffing Oscillator}
We consider chaotic dynamics of a 2D duffing oscillator in \citet{anderson2024fisher}
\begin{align*}
    dx_1 &= x_2 dt \\
    dx_2 &= (1.0x_1 -0.2 x_2 -1.0 x_1^3)dt + \frac{1}{\sqrt{20}} dw.
\end{align*}
The initial distribution is multivariate Gaussian
$$
p_0(x)\sim \mathcal{N}(\mu, \Sigma).
$$
The computation domain is $t \in [0,2.5]$ and $x_1,x_2\in[-2,2]$, and the system parameters are $\mu=[1,1]^T, \Sigma=diag([0.05,0.05])$.
Again, extensive Monte-Carlo simulations using Euler Scheme are carried out to estimate the "true" PDF $p(x,t)$, with integration time step $\Delta t=0.005$ and $2 \times 10^7$ samples.

\subsection{High-Dimensional Time-Varying OU}\label{appendx:HighD-sys}
The generic dynamics of high-dimensional time-varying Ornstein-Uhlenbech is
$$
dx=\Big( A_n+\Delta A_n(t)x \Big)dt+B_n dw,
$$
where $x,w \in R^n$. The initial distribution is multivariate Gaussian
$$
p(x,0)\sim \mathcal{N}(\mu_n, \Sigma_n).
$$
Since there is no closed-form solution as discussed in Sec. XXX, we use Euler forward numerical integration with $\Delta t=0.0001$ time step to obtain the "true" PDF. The computation domain is $t\in[0,1]$ and $x\in[-1,1]^n$.
Below, we summarize the system parameters of 3D, 7D, and 10D experiments. For $n=3$, we set
\begin{align*}
    & A_3 = diag([0.3, 0.3, 0.3]) \\
    & \Delta A_3(t) = \sin(t)\begin{bmatrix}
        0 & 0 & -0.1 \\
        0 & 0 & 0 \\
        0 & 0 & 0 
    \end{bmatrix} \\
    & B_3 = diag([0.05,0.05,0.05]) \\
    & \mu_3 = [-0.2, 0.2, 0.0]^T \\
    & \Sigma_3 = diag([0.1, 0.1, 0.1]).
\end{align*}
For $n=7$, we set
\begin{align*}
    & A_7 = \begin{bmatrix}
        0.3 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0.3 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0.15 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0.3 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0.3 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & -0.3 & 0 \\
        -0.01 & 0 & 0 & 0 & 0 & 0 & 0.3 \\
    \end{bmatrix} \\
    & \Delta A_7(t) = \cos(t)\begin{bmatrix}
        0 & 0.1 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0.1 & 0.2 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & -0.1 & 0 & 0 & 0 & 0 & 0 \\
    \end{bmatrix} \\
    & B_7 = diag([0.05,0.05,0.05,0.05,0.05,0.05]) \\
    & \mu_7 = [0,0,0,0,0,0,0]^T \\
    & \Sigma_7 = diag([0.12,0.12,0.12,0.12,0.12,0.12,0.12]).
\end{align*}
For $n=10$, we set
\begin{align*}
    & A_{10} = \begin{bmatrix}
        0.3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0.3 & 0 & 0 & 0 & 0.03 & 0 & 0 & 0 & 0 \\
        0 & 0 & -0.3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0.3 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0.06 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0.3 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0.3 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.21 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.3 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & -0.02 & 0 & 0.3 \\
    \end{bmatrix} \\
    & \Delta A_{10}(t) = \sin(t)\begin{bmatrix}
        0 & 0.1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0.05 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
        0 & -0.1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
    \end{bmatrix} \\
    & B_{10} = diag([0.05,0.05,0.05,0.05,0.05,0.05,0.05,0.05]) \\
    & \mu_{10} = [0,0,0,0,0,0,0,0,0,0]^T \\
    & \Sigma_{10} = diag([0.12,0.12,0.12,0.12,0.12,0.12,0.12,0.12,0.12,0.12]).
\end{align*}

\subsection{1D Heat}\label{appendix:1D-HEAT-sys}
We consider a one-dimensional heat equation as in Appendix~\ref{proof:1D_Heat}:
\begin{align*}
    u_t-u_{xx}=0,
\end{align*}
subject to initial and Dirichlet boundary conditions
\begin{align*}
& u_0(x)=-\sin(\pi x), \\
& u(\pm1,t)=0,\forall t.
\end{align*}
The computation domain is $t \in [0,1]$ and $x \in [-1,1]$.
For this particular setting, analytical solution exists
$$ u(x,t)=-\sin(\pi x)\exp^{-\pi^2t}.$$