## Appendix A Projection Algorithm for Physically Constrained Particle-Distribution Prediction
Finally, the number density function (\(\rho_{\text{target}}\)) is constructed (Equation (14)). \[\text{ln}(\rho)=\text{ln}(\text{A})-\text{BR} \tag{13}\]  
\[\rho_{\text{target}}(\text{R})=\text{A}_{\text{fit}}\text{e}^{-\text{BR}} \tag{14}\]  
Figure 14 illustrates the results for a specific test sample (with 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%). Although the target number density function--derived via the linear regression of the logarithmically transformed density against the radial distance--captures the general trend of decay from the center outward, noise and local prediction errors in the actual data may cause the fitted function to inadequately represent the maximum density expectedin the central region. Consequently, to ensure that the candidate point selection strictly conforms to the physical prior (i.e., the central region should inherently exhibit a higher particle number density), an additional central bias factor, C, is introduced (Equation (A10)). This factor adjusts the weights of candidate points based on their radial distances, thereby amplifying the influence of those nearer to the center. In turn, during the local reallocation process, this central bias compensates for any discrepancies in the fitted function, ultimately yielding final predictions that more accurately reflect the true particle distribution. \[\mathrm{C(R_{i})}=\left(\frac{R_{\mathrm{max}}-R_{i}}{R_{\mathrm{max}}} \right)^{\gamma}\] (A10)  
where \(R_{i}\) is the radial distance of a candidate point, \(R_{\mathrm{max}}\) is the maximum radial distance, and \(\gamma\) (=2) is the exponent for the central bias factor. Ultimately, the initial weight assigned to each candidate point in the under-estimated region is defined as the product of the target density function--obtained through fitting--and the central bias factor (Equation (A11)). \[\rho_{c,\text{log}}=\log_{10}(\rho_{c}+1)\]0 (A11)  
Subsequently, the initial weight of each candidate point is normalized to yield the final weight  
\[\rho_{c,\text{log}}=\log_{10}(\rho_{c}+1)\]1 (A12)  
where N is the total number of invalid points. After computing and normalizing candidate point weights, a local reallocation strategy is applied to relocate invalid points from the dead zone into regions with under-estimated number density. For each invalid prediction within the dead zone, a KD-tree identifies a set of candidate locations in its local neighborhood, each assigned weights derived from the target density function and central bias. A probabilistic sampling approach then selects a candidate based on its normalized weight \(w_{i}\), conferring higher selection probability to points with larger weights. The chosen candidate's coordinates replace the invalid prediction, thereby relocating the point from the dead zone to a physically plausible region. This approach relocates each invalid point to a position more consistent with physical constraints. These corrected predictions are then combined with the valid ones to produce the final output set. By using a localized, weight-based probabilistic sampling strategy, the method effectively adjusts invalid points within the dead zone, preserving physical continuity and aligning the final predictions with the particle distribution seen in the training data.