## 6 Discussion and Conclusions  
In this study, we developed a systematic approach (outlined in Figure 1) for bridging two-dimensional continuous models of diffusive transport in the presence of rigid internal obstacles with perfectly elastic collisions with two types of surrogate models. The first type is a lattice-based surrogate model that exploits symmetries in the RVE of the continuous domain used for direct numerical simulation. The second type is a square lattice surrogate model that can enable the accurate estimation of diffusivities to be subsequently used in more general lattice-based modeling frameworks where diffusion is one of several important coupled mechanisms. Here, diffusion is often the fastest spatiotemporal mechanism. Our systematic approach can be used, first, to establish the discrete model probabilities and grid size for building 2D lattice models in applications that couple diffusion with additional mechanisms in a background medium having specific geometric properties. Conversely, in other applications, the ability to calibrate the grid size such that the associated probabilities are consistent with an observed or prescribed diffusivity may also be beneficial.  
While the level of noise and error increases in the more complex (multisize obstacles) model considered in this study, it remains in a range that is comparable to the level of noise or variation typical for 2D lattice-based simulations with underlying stochastic or probabilistic mechanisms. Our approach is versatile in the sense that it can be extended to other geometric configurations of obstacles in an RVE exhibiting several degrees of symmetry. Similarly, the techniques used in our overall approach can be extended to three-dimensional lattice-based models for materials with internal geometric symmetries. Nevertheless, it should be noted that the computational time needed to achieve stationarity of the underlying continuous and lattice-based surrogate model probabilities can increase substantially with increasing model complexity. Consequently, the extent to which computation time is practical (versus prohibitive) will depend on the degree of internal symmetry in the background medium as well as availability of ultra-high-performance computing resources.  
Conceptualization, T.M.A. and M.A.H.; Methodology, T.M.A. and M.A.H.; Validation, T.M.A. and M.A.H.; Writing--original draft, T.M.A. and M.A.H.; Writing--review & editing, T.M.A. and M.A.H.; Visualization, T.M.A. and M.A.H. All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.  
This research was funded and the work was supported by the US National Science Foundation via grant DMS-1638521 and an NSF Graduate Research Fellowship.  
Not applicable.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(R_{\text{obs}}\) & \(q_{y}=2p_{2}\) & \(q_{x}=p_{3}+p_{4}\) & \(\Delta x_{M}\) (\(\kappa_{V}\) est.) & \(\Delta x_{M}\) (\(\kappa_{H}\) est.) \\ \hline
0.5 & 0.0037360 & 0.0037629 & 4.2097 & 4.1652 \\
1.0 & 0.0035430 & 0.0035489 & 4.0917 & 4.1101 \\
1.5 & 0.0033788 & 0.0033849 & 4.3173 & 4.1280 \\
2.0 & 0.0032406 & 0.0032431 & 4.4659 & 4.3340 \\
2.5 & 0.0034196 & 0.0034172 & 4.1858 & 3.9538 \\
3.0 & 0.0033004 & 0.0033094 & 4.0902 & 3.7193 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Estimated values of grid size \(\Delta x_{M}\) based on the lattice-based surrogate model simulation results for the multisize case using the effective horizontal diffusivity (\(\kappa_{H}\)) and the effective vertical diffusivity (\(\kappa_{V}\)). The resulting transition (leave) probabilities in the square lattice surrogate model are also shown.  
**Data Availability Statement:** The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. The data presented in this study are available on request from the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.