## 2 Methodology
### The 1 st ML Model to Predict the Spatial Distribution of Particles on the Substrate
#### 2.3.2 Mathematical Transformation--Symbolic Regression  
Traditional machine learning models often operate under inherent assumptions about the data structure [24]. For instance, linear regression assumes a linear relationship between features and the target variable. While its flexibility can be extended by introducing polynomial or interaction terms, selecting the appropriate transformations still requires manual prior knowledge. Similarly, decision trees, despite their ability to automatically partition the feature space, rely on a series of local decisions based on discrete split points [25]. As a result, when the underlying data follows a smooth nonlinear mapping, decision trees may require a large number of splits to approximate it effectively, which can lead to overfitting [25]. However, applying suitable mathematical transformations--such as polynomial features, logarithmic scaling, or kernel methods--before modeling can project the data into a more structured space, enabling the model to better capture the underlying patterns [26]. For instance, neural networks automatically learn optimal feature mappings through their layered structure [27]. Each layer applies nonlinear transformations to the input, progressively projecting the data into higher-dimensional feature spaces where complex patterns become more separable. This hierarchical feature extraction allows neural networks to capture intricate relationships that may not be evident in the raw data. However, this process typically requires large amounts of data because deep networks contain many parameters, and without sufficient training examples, they risk overfitting rather than learning meaningful transformations. To address these challenges while drawing inspiration from neural networks, symbolic regression is employed to automatically identify the most effective mathematical transformations for the data. More specifically, symbolic regression explores a wide range of functional forms--such as logarithmic, exponential, or polynomial relationships--without assuming a predefined structure. By discovering optimal transformations, it projects the data into a space where decision boundaries become clearer, improving model expressiveness while reducing reliance on manual feature engineering. This adaptability makes symbolic regression particularly valuable for datasets with unknown or complex relationships. In symbolic regression, a \"symbolic tree\" is a tree-structured representation of mathematical transformations. Each node in the tree corresponds to an operator or operand. For instance, unary operators (e.g., _sin_, _cos_, and _log_) take a single child node as input, while binary operators (e.g., +, -, \(\times\), and /) require two child nodes, typically left and right. Leaf nodes usually represent feature variables or constants. By combining these operators at different levels of the tree, each symbolic tree can be parsed into a corresponding mathematical transformation. An example of a symbolic tree is shown in Figure 12.  
Figure 11: Dataset partitioning for model training. In the current multi-feature symbolic regression framework, each candidate solution in the population consists of multiple symbolic trees--one per input feature. Specifically, since the input comprises ten features, each candidate solution is represented by ten distinct symbolic trees, each encoding a unique mathematical transformation (e.g., using operators such as addition, subtraction, sine, logarithm, etc.). This design allows the algorithm to discover feature-specific expressions, ensuring that each feature is transformed independently by its own tree. The complete workflow of the symbolic regression process is illustrated in Figure 13. During the population initialization phase, symbolic trees are randomly generated for each feature according to predefined constraints, such as maximum tree depth and the set of allowable operators.