## 2 Methodology
### Sampling  
Typically, each CFD simulation contains hundreds of thousands of data points. In our previous study [13], a single simulation included nearly a million data points representing particles trapped on the substrate, making it impractical to use the raw simulation results directly for machine learning due to high computational demands. To overcome this challenge, we propose a sampling method that combines feature stratification with greedy sampling to extract representative subsets from the large dataset. Importantly, the data obtained through this sampling approach is used in both models, ensuring that each model benefits from a consistent and representative dataset while significantly reducing computational resource consumption. Firstly, the CFD results indicate that axial powder injection and higher gas velocity along the jet centerline cause most particles to remain near the centerline (see the particle distribution in Figure 1, where the point (\(y=0\), \(z=0\)) represents the jet centerline), gaining significant thrust and momentum in this region. As a result, random sampling could disproportionately select points from the central region (i.e., near the jet centerline), leading to oversampling there and undersampling in other regions. To mitigate this, stratified sampling based on the particles' \(y\)- and \(z\)-coordinates is adopted to ensure balanced representation across different regions. Specifically, particles are initially sorted in ascending order based on their \(y\)-coordinate values (feature \(y\)). \[ \begin{split}&\text{{[$y_{1},y_{2},y_{3}\ldots,y_{N}$]}}\\ &\text{{y}}_{N-1}<\text{{y}}_{N}\end{split}  \tag{5}\]To ensure an even distribution of particles within each interval in terms of horizontal coordinate, a quantile-based partitioning method is employed. \[ \begin{array}{c}q_{k}=\frac{k}{y_{y}}\\ k=1,2,\ldots,b_{y}-1\end{array}  \tag{6}\]  
where \(q_{k}\) is the \(k\)-th quantile of \(y\) and \(b_{y}\) is the number of intervals. Based on the quantiles, the particles sorted by their \(y\)-coordinates are divided into \(b_{y}\) intervals. Each interval contains an approximately equal number of particles, ensuring uniform distribution within each \(y\)-coordinate range. This partitioning ensures that the stratified sampling captures the characteristics of the data across the entire \(y\)-coordinate spectrum without overrepresenting or underrepresenting any specific region. \[ \begin{array}{c}[a_{0},a_{1}),[a_{1},a_{2}),\ldots\left[a_{b_{y}-1},a_{b_{y} }\right]\\ a_{0}=\min(y),\ a_{b_{y}}=\max(y)\end{array}  \tag{7}\]  
For each \(k\left(=1,2,  
particles exceeds the assigned sample size, a greedy algorithm is applied to determine the samples. A schematic of the stratified and greedy sampling process is shown in Figure 3.  
The purpose of the greedy algorithm is to select data points from the layer that are as diverse or dispersed as possible within the feature space. The algorithm generally follows three steps: (1) randomly select a seed point and add it to the sample set; (2) standardize all features to eliminate scale differences; (3) iteratively choose the candidate point that maximizes its minimum Euclidean distance from the current sample set. As mentioned, first, a point is randomly selected as the seed point \(x\) and added to the sample set \(C\). Then, in each iteration, the algorithm evaluates all the unselected candidate points. Specifically, for each candidate, it computes its Euclidean distance from the current sample set \(C\); that is, it calculates the distances between the candidate and every point in the sample set and takes the minimum of these distances (i.e., the distance from the candidate to its nearest sample) (Equations (12) and (13)). This minimum value serves as the criterion to assess the candidate's remoteness from the existing samples.