## 2 Methodology
### The 1 st ML Model to Predict the Spatial Distribution of Particles on the Substrate
#### 2.3.1 Interpolation  
A feature removal method was employed to evaluate the contribution of individual features to the model's performance with 40 groups used as the training set and 8 groups as validation. The specific steps involved sequentially removing each input feature, retraining the model, and calculating the extent of performance degradation. The importance of each feature was quantified as a percentage relative to the baseline R\({}^{2}\) value. To enhance the reliability of the evaluation, multiple regression models, such as decision trees, linear regression, and multilayer perceptrons, were used for the experiments. The average R\({}^{2}\) value across these models was taken as the final measure. The results are presented in Figure 9.  
The results indicate that two inlet condition features (gas temperature and pressure) negatively impacted the prediction performance. However, from a practical perspective, the inlet condition remains the most strongly correlated variable with particle impact characteristics at the substrate [22]. This suggests that the negative impact arises due to insufficient data, preventing the model from effectively learning the relationship between the inlet condition and particle impact characteristics [23]. More specifically, the dataset contains only two distinct inlet condition configurations [13], resulting in a severe lack of samples. This scarcity makes it difficult for the model to extract meaningful information, leading it to treat these features as noise. This also explains why removing these features improved the model's performance. To address this issue, an interpolation method is planned to expand the database along key parameter directions, enhancing the model's ability to learn from features with limited variability and better capture their potential correlations. Specifically, the interpolation is applied to key parameters such as mask properties (i.e., wire diameter, opening size, and open area), standoff distances (e.g., distances from the nozzle exit to the substrate and from the nozzle exit to the mask), and inlet conditions (e.g., gas inlet pressure and temperature). This process will generate a richer set of data samples. One critical assumption is that while varying the feature under consideration, all the other features remain constant. Figure 9: Feature importance analysis using the “dropping feature” method, where the decrease in R\({}^{2}\) indicates the impact of removing each feature. The detailed interpolation and augmentation workflow is documented in Appendix B. To facilitate comparison, an additional feature removal analysis was conducted after incorporating interpolated data into the training set. Previously, the dataset consisted of 40 groups for training and 8 groups for validation. After adding 86 interpolated data points, the training set expanded to 40 + 86 groups, while the validation set remained at 8 groups. This adjustment aimed to evaluate how the inclusion of approximated values influenced the model's ability to capture feature dependencies and contribute to prediction accuracy. With additional interpolated data, the model demonstrated an improved capacity to learn the relationships between the input features and the target variables, leading to an overall increase in feature importance for most variables. This means that the expanded dataset provided more context, enabling the model to establish more meaningful dependencies. However, substrate SOD remains an exception, showing only a 2.31% R\({}^{2}\) increase, indicating that its impact on predictions is still minimal. This suggests that while interpolation has enhanced the relevance of other features, substrate SOD may require further investigation.