## 2 Materials and Methods
### Hyperparameters Optimization  
In this study, a systematic strategy is employed for hyperparameter tuning, which focuses on refining machine learning models via the GridSearchCV technique. Specifically, GridSearchCV is utilized across GBR, XGBoost, SVR, ANN, ANN-GMDH RFR and DTR, models. This approach involves systematically exploring a predefined grid of hyperparameter values, coupled with cross-validation for each combination, to identify the most effective set that enhances model performance. Considering the more compact and manageable hyperparameter spaces of these models, a comprehensive exploration proves both feasible and advantageous [99; 100; 101]. While acknowledging the larger and intricate hyperparameter spaces of the GBR and ANN models, requiring additional optimization time, the study opts for GridSearchCV for consistency. This strategic choice guarantees  
Figure 11: Flowchart of symbolic regression process.  
a systematic and exhaustive examination of the hyperparameter space, aligning with the study's methodology and obviating the necessity for Randomized Search CV [101].  
Artificial neural networks (ANNs) powered by TensorFlow are instrumental in unraveling complex patterns within datasets. Specifically designed for regression tasks, ANNs utilize a structure of interconnected nodes organized into layers to establish correlations between input features and continuous output values. In TensorFlow-based ANNs, hyperparameter optimization involves thorough exploration of the parameter space, encompassing architecture parameters like activation functions, neuron count per layer, and layer count, as well as training process parameters such as number of training epochs, batch size, and learning rate. This systematic approach aims to identify the optimal combination of hyperparameters that enhances predictive accuracy, exploring various configurations and leveraging TensorFlow's capabilities for efficient evaluation and training.  
This process entails a thorough exploration of these models' parameter spaces to identify optimal configurations. The optimal settings for each model regarding \(q_{u}\) and \(E_{t}\), respectively, are displayed in Tables 5 and 6.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Hyperparameter** & **Value** \\ \hline \multirow{3}{*}{DTR} & max\_depth & 10 \\  & min\_samples\_leaf & 1 \\  & min\_samples\_split & 1 \\ \hline \multirow{3}{*}{RFR} & max\_depth & 10 \\  & min\_samples\_leaf & 1 \\  & min\_samples\_split & 1 \\  & n\_estimators & 130 \\ \hline \multirow{3}{*}{GBR} & subsample & 0.8 \\  & n\_estimators & 100 \\  & min\_samples\_split & 12 \\  & min\_samples\_leaf & 2 \\  & max\_depth & 3 \\  & learning\_rate & 0.6 \\ \hline \multirow{3}{*}{XGBoost} & subsample\_bytes & 0.8 \\  & learning\_rate & 0.4 \\ \cline{1-1}  & max\_depth & 3 \\ \cline{1-1}  & n\_estimators & 200 \\ \cline{1-1}  & subsample & 0.6 \\ \hline \multirow{3}{*}{SVR} & C & 100,000 \\  & epsilon & 10 \\ \cline{1-1}  & kernel & rbf \\ \hline \multirow{3}{*}{ANN} & number of neurons in layer1 & 10 \\  & number of neurons in layer2 & 5 \\  & learning\_rate & 0.05 \\ \cline{1-1}  & batch size & 10 \\ \cline{1-1}  & hidden layers function & ReLU \\ \cline{1-1}  & linkage between the hidden \\ \cline{1-1}  & layer and the ultimate output & ReLU \\ \hline \multirow{3}{*}{ANN-GMDDH} & number of neurons in layer1 & 20 \\  & number of neurons in layer2 & 30 \\ \cline{1-1}  & learning\_rate & 0.09 \\ \cline{1-1}  & batch size & 14 \\ \cline{1-1}  & hidden layers function & ReLU \\ \cline{1-1}  & linkage between the hidden \\ \cline{1-1}  & layer and the ultimate output & ReLU \\ \cline{1-1}  & layer function & \\ \hline \hline \end{tabular}
\end{table}
Table 5: Optimal hyperparameters for models predicting ‘\(q_{u}\)’.