## 2 Materials and Methods  
### Symbolic Regression (SR)  
Symbolic regression (SR) is a robust machine learning approach that extracts mathematical equations directly from data, providing a distinctive blend of flexibility and interpretability. Unlike traditional neural networks (NNs), which identify relationships between input and output data by minimizing Mean Square Error (MSE), SR delivers explicit analytical expressions for these relationships, overcoming the limitations of NNs. NNs, often regarded as \"black-box\" models, obscure the underlying input-output relationships and require passing new inputs through the model, which can be computationally expensive and impractical for real-time applications. By merging the adaptability of NNs with the clarity of analytical models, SR generates a mathematical formula representing the discovered relationship at the end of training. This eliminates the need for the complex model during deployment, significantly reducing computational overhead and making SR particularly suitable for real-time applications, such as dynamically adjusting coil distances in Wireless Power Transfer (WPT) systems to optimize power transfer efficiency. The SR process employs genetic algorithms to search for the optimal formula \(f_{SR}\) that best satisfies the relationship output = \(f_{SR}\)(input). The search space includes predefined operators, such as unary operators (e.g., exponential, sine, and cosine) and binary operators (e.g., addition, multiplication, and power), which act as the \"genes\" in the genetic algorithm. Each candidate formula in the population represents a unique combination of these operators. The genetic algorithm evaluates candidate formulas based on two key criteria: accuracy (minimizing MSE) and complexity (favoring simpler expressions). Formulas with lower MSE and reduced complexity are more likely to survive and propagate to subsequent generations through mechanisms like mutation, which alters constant values or replaces operators, and reproduction, which combines elements from two parent formulas to create new offspring. This iterative process continues over multiple generations until convergence is achieved, yielding a final formula that balances accuracy and simplicity. Thus, the ability of SR to produce explicit mathematical expressions not only enhances transparency by revealing underlying data relationships but also improves efficiency by enabling the direct application of the derived formula without complex computations. This makes SR ideal for systems requiring rapid decision-making, such as adaptive WPT systems, while demonstrating its potential to enhance both performance and understanding in engineering domains. Despite its advantages, SR has certain limitations. Due to the vast research space, the method can be computationally slow. Additionally, like other machine learning techniques, SR is prone to overfitting, where the formula may fit the data well but fail to capture the underlying physics. A notable solution to this issue was proposed by [40], which leverages symmetries and simplifications often used by physicists to reduce the search space and identify the simplest formula that best fits the data. Furthermore, SR performs well primarily with small datasets. However, techniques combining deep neural networks with SR have been developed to extend its applicability to larger datasets.