## 4 Result and Discussion  
### Model Training  
In this study, a novel meta-heuristic optimization algorithm, WAOA, is combined with ANN, RF, and XGBoost to construct more efficient hybrid multi-output models for predicting borehole breakout size. According to the experience and theoretical research of researchers on hyperparameter adjustment of ANN, RF and XGBoost regression prediction model with multi-feature input, the hyperparameter objects and ranges optimized by WAOA in this study are shown in Table 3.  
Figure 7 illustrates the construction, optimization and comparison process of the hybrid multi-output model proposed in this study. After determining the influencing factors of borehole breakout size by analyzing and comparing the existing literature, a substantial amount of relevant data was collected from the existing literature to construct a dataset. To balance the accuracy requirement of the model to explore the intricate nonlinear relationship between input and output factors in the training process, and the rigor requirement of the model comparison process, the dataset was randomly divided into the training set (80%) and the test set (20%). To comprehensively investigate the intricate nonlinear relationship between input features and borehole breakout size, and to leverage the potential correlation between breakout size, the ANN, RF, and XGBoost was employed as the foundation, with WAOA utilized as the optimization tool to develop a hybrid multi-output prediction model for borehole breakout size. In the optimization process of WAOA, MSE is chosen as the fitness evaluation metric which can be calculated as:  
\[MSE=\frac{\sum\limits_{i=1}^{n}{\left({f_{i}-y_{i}}\right)^{2}}}{n} \tag{13}\]  
Simultaneously, to address inconsistencies in units among different outputs and ensure a balanced consideration of various output prediction accuracies in the overall fitness evaluation of the model, all input and output features were normalized to [0, 1]. The normalization process can be presented as:  
\[x=\frac{{x^{\prime}-x_{\min}}}{{x_{\max}-x_{\min}}} \tag{14}\]  
where \(x\) denotes the value after normalization, \(x^{\prime}\) denotes the value before normalization, \(x_{\min}\) denotes the minimum value of the feature, \(x_{\max}\) denotes the maximum value of the feature. This normalization is an equiproportional transformation of the entire range of \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Hyperparameters** & **Range** & **Parameter Meaning** \\ \hline \multirow{4}{*}{ANN} & Learning rate & [0.00001, 0.1] & Learning rate of the optimizer \\ \cline{2-4}  & Num dense layers & [1; 10] & Number of hidden layers \\ \cline{2-4}  & Num dense nodes & [5; 110] & Number of nodes for each layer \\ \cline{2-4}  & Activation & Relu & Activation function \\ \hline \multirow{2}{*}{RF} & n\_estimators & [1; 1100] & The number of trees \\ \cline{2-4}  & max\_depth & [10; 1100] & The maximum tree depth \\ \hline \multirow{3}{*}{XGBoost} & max\_depth & [1; 10] & The maximum tree depth \\ \cline{2-4}  & learning\_rate & (0, 1] & Rate of iteration \\ \cline{1-1} \cline{2-4}  & n\_estimators & [1; 500] & The number of trees \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Hyperparameter settings and optimization ranges. features, it does not change the distribution shape of the features, and therefore does not disrupt the potential nonlinear relationships in the dataset. Since model hyperparameter optimization involves continuously adjusting the hyperparameter settings to fit the training data, overfitting is prone to occur if a single training set is used exclusively for training and evaluating the fitness of adjustments. To mitigate the risk of overfitting in the iterative process of the hybrid multi-output model, 5-fold cross-validation was employed during model training. Multifold cross-validation is an effective method for preventing overfitting during iteration and enhancing model generalization performance.