## 3 Models
### Multisize Obstacles Model
#### 3.3.2 Lattice-Based Surrogate Model  
Figure 5 illustrates the construction of the 2D lattice-based surrogate model for the multisize obstacles case. A periodic extension of the RVE is shown, with primary transitions from subdomain 1 (top-left corner) illustrated. Due to inherent symmetries, all subdomains have the same number of distinct transition types. There are four types of transition with corresponding probabilities (Figure 5b). In type 1, the particle stays in its current subdomain (\(p_{1}^{M}\)). In types 2-4, it transitions to a neighboring subdomain (2,3 or 4) having two obstacles. We delineate the three latter types into probabilities as follows: the two obstacles are of different sizes (\(R_{\text{obs}_{1}}\), \(R_{\text{obs}_{2}}\)) along the boundary (\(p_{2}^{M}\)); the two obstacles are the same size \(R_{\text{obs}_{2}}\) along the boundary (\(p_{3}^{M}\)); or the two obstacles are the same size \(R_{\text{obs}_{1}}\) along the boundary (\(p_{4}^{M}\)). Again, the probabilities must sum to one; the three surrogate model probabilities need to be calculated from the results of the continuous model simulations.  
Again, the time course \(\eta(t_{k})\) records the state of the particle as the index of the \(i\)th subdomain (\(i=1,...,4\)) at time \(t_{k}\) if its center is located in that subdomain at time \(t_{k}\). For the multisize obstacles case, we differentiate between a horizontal transition within the RVE versus a horizontal transition into an extended domain by introducing an additional time course \(\zeta(t_{k})\) (\(t_{k}=k\Delta t,\ k=0,\dots,N\)), where  
\[\zeta(t_{k})=\begin{cases}1&\text{if a horizontal transition occurred into an extended domain}\\ 0&\text{otherwise (includes all vertical transitions)}.\end{cases}. \tag{15}\]  
In this case, the transition matrix \(S^{(k)}\)(for time steps \(k=1,\dots,N\)) is a \(4\times 5\) (augmented) matrix; the fifth column tracks horizontal transitions occurring when the particle moves into an extended domain. For a large majority of time steps, \(\zeta(t_{k})=0\) and \(S^{(k)}\) are updated in a similar manner as in the equally spaced obstacles case via Equation (12) (Section 3.2.2). Alternatively, when \(\zeta(t_{k})=1\), only the new column in the augmented matrix is updated via the formula  
\[S^{(k)}(\eta(t_{k-1}),5)=S^{(k)}(\eta(t_{k-1}),5)+1,\ \ k=1,...,N. \tag{16}\]  
and \(S^{(k)}\) is then used to build the vector \(\vec{p}^{M}\) of five transition probabilities.  
Within each time step and for each of the four subdomains, a vector \(\vec{T}_{i}^{M}\) (\(i=1,2,3,4\)) is created to count transitions, according to the following rules: (i) the first component counts the number of times a particle stayed in its current subdomain; (ii) the second component counts the number of times a particle moved vertically; (iii) the third component counts the number of times a particle moved horizontally but remained in the original RVE; (iv) the  
Figure 5: Building a lattice-based surrogate model for the continuous model in the multisize obstacles case: (**a**) four subdomains are used to exploit symmetries of the obstacle arrangement, and transitions between subdomains are tracked. Smaller green circles indicate obstacles of fixed radius \(R_{\text{obs}_{1}}=1.0\), and larger blue circles indicate obstacles with varying obstacle radius \(R_{\text{obs}_{2}}\); (**b**) lattice-based surrogate representation with red circles denoting corresponding subdomains in the continuous model.  
fourth component counts the number of times a particle moved horizontally but moved to the extended domain.  
Due to the symmetries in this case, we create the transition probability vector \(\vec{p}^{M}\) via the average:  
\[\vec{p}^{M}=\frac{1}{N}\sum_{j=1}^{N}\vec{T}_{j}^{M}. \tag{17}\]