## Appendix A Projection Algorithm for Physically Constrained Particle-Distribution Prediction  
Initially, a detection algorithm is employed to identify all predicted points located within the dead zone (see Figure 8), which are then isolated and stored as a separate set for subsequent reallocation. The algorithm used in this step is identical to the soft filter algorithm applied in the preceding submodule. It is theoretically expected that, in the absence of mask interference, the deposited single-track profile or the number density of particles on the substrate should peak at the center (\(y=0\), \(z=0\)) and decrease continuously outward [36]. If a sudden decline or an abrupt transition to zero in the track profile or the number density function is observed within the deposition region, the presence of an under-estimation issue can be inferred. To address this, the model utilizes a heatmap constructed from the number density of particles to precisely locate regions of abrupt decline, thereby delineating the under-estimated areas. To generate the heatmap, first, the entire 5 mm \(\times\) 5 mm region (a quarter of the substrate shown in Figure 1) is divided into a 2000 \(\times\) 2000 grid. Next, each grid point is evaluated: if the point lies within the dead zone (as determined by physical priors), its number density is directly set to zero; otherwise, the local number density is computed by counting the number of predicted points (N) within a neighborhood of radius R\({}_{c}\) (Equation (A1)). To expedite the construction of the density map, a KD (k-dimensional) tree is employed for efficient neighborhood querying. Furthermore, to better accommodate the wide range of number density values, a smoothing transformation is applied to the data (Equation (A2)). The constant 1 in Equation (A2) is added to the number density values to prevent the logarithm from yielding negative infinity in regions where the number density is zero. Figure A1 presents the logarithmic density heatmap for a specific test sample (an inlet pressure of 4 MPa, an inlet temperature of 800 \({}^{\circ}\)C, a substrate SOD of 20 mm, a mask SOD of 16 mm, a mask wire diameter of 0.46 mm, a mask opening size of 1.14 mm, and an open area of 51%). \[\rho_{c}(y,z)=\frac{\text{N}}{\pi{\text{R}_{c}}^{2}}\] (A1)  
\[\rho_{c,\text{log}}=\log_{10}(\rho_{c}+1)\] (A2)Subsequently, in regions outside the dead zone, effective areas are identified to correct the under-estimation issue, where regions expected to exhibit high number density are mistakenly predicted as low number density (see Figure A1). It is noteworthy that in Figure A1, the data have been subjected to a logarithmic transformation to promote overall smoothness; thus, any pronounced discontinuities observed in the log-density map can be reliably attributed to under-estimation. To address this issue, first, the logarithmically transformed number density data are processed using the Otsu algorithm to automatically determine an optimal threshold [37]. The Otsu algorithm works by iteratively evaluating candidate thresholds: for each candidate, the data are divided into two groups (below and above the threshold), and the weight of each group is calculated as the proportion of data points in that group. The mean value for each group is computed, and then the between-class variance is calculated using  
\[\sigma_{b}^{2}=w_{0}\,w_{1}\left(\mu_{0}-\mu_{1}\right)^{2} \tag{1}\]  
where \(w_{0}\) and \(w_{1}\) are the weights (proportions), and \(\mu_{0}\) and \(\mu_{1}\) are the means of the two groups, corresponding to low-number-density area and high-number-density area, respectively. The candidate threshold that maximizes the between-class variance is selected as the optimal threshold; moreover, a randomized search has been applied to further refine this selection. Regions with number density values exceeding this Otsu threshold are designated as high-number-density areas, while those with values falling below the threshold are labeled as low-number-density areas.