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\section{Empirical Simulations}
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\subsection{Generation of 2D synthetic data}

We first employ the reflected SB algorithm to generate three synthetic examples: checkerboard and Gaussian mixtures from a Gaussian prior and spiral from a moon prior. The domains are defined to be flower, octagon, and heart, where all boundary points are defined to have proper unit-vectors. We follow \citet{forward_backward_SDE} and adopt a U-net to model $(\textcolor{red}{\overrightarrow\bz_t^{\theta}}, \textcolor{teal}{\overleftarrow\bz_t^{\omega}})$. %We include position encoding and train with a AdamW optimizer. 
We chose RVP-SDE as the base simulator from time $0$ to $T=1$, where the dynamics are discretized into 100 steps. % using the Euler-Maruyama scheme.



Our generated examples are presented in Figure \ref{checkboard} and \ref{rsb_3_domains}. We see that all the data are generated smoothly from the prior and the forward and backward process matches with each other elegantly. To the best of our knowledge, this is the first algorithm (with OT guarantees) that works on custom domains. Other related work, such as \citet{reflected_diffusion_model}, mainly focuses on hypercubes in computer vision. We also visualize the forward-backward policies $\textcolor{teal}{\overleftarrow \bz_t^{\omega}}$ 
 and $\textcolor{red}{\overrightarrow\bz_t^{\theta}}$ in Figure \ref{rsb_3_domains}. Our observations reveal that the forward vector fields $\textcolor{red}{\overrightarrow\bz_t^{\theta}}$ demonstrate substantial nonlinearity when compared to the linear forward policy in SGMs, and furthermore, the forward vector fields exhibit pronounced dissimilarity when compared to the backward vector fields $\textcolor{teal}{\overleftarrow \bz_t^{\omega}}$.






