## 4 Result and Discussion
### Model Comparison  
Through comparing the optimization effects of the optimization algorithms on the multi-output ANN, RF, and XGBoost models across various population size settings, the optimal results of the three hybrid multi-output models were identified. Regarding the predictive performance, all three models exhibited exceptionally strong capabilities. It is clear that they accurately captured the potential relationship between input features and outputs. However, it remains uncertain whether the potential relationship between the outputs has been appropriately utilized. To further select the final recommended hybrid multi-output model and validate the effectiveness of employing a multi-output setup, a comparative analysis of the obtained models is conducted. Figure 10 provides an intuitive representation of the effectiveness of three different hybrid multi-output models in predicting borehole breakout size. In the scatter plots, where the actual values are plotted on the \(x\)-axis and the predicted values on the \(y\)-axis, the scatter points in all figures closely align around the \(y=x\) line, with most data points exhibiting a relative error within 15%. From the kernel density distributions of the actual and predicted values in the upper and right parts of the scatterplot, the distributions of predicted and actual values are very similar in both the training and test sets. This evidences the excellent performance of the three hybrid multi-output models, with accurate exploration of the potential correlation between input and output. It is noteworthy that there is no significant difference in the predictive performance of the three models, either on the training or test sets, concerning predicting \(\theta\) and \(r_{b}/r\). However, regarding the difference in prediction performance between the training set and the test set, the WAOA-ANN model demonstrates a more stable performance, whereas the WAOA-RF and WAOA-XGBoost models exhibit a noticeable decrease in predictive performance in the test set compared to the training set. When comparing the three hybrid multi-output models, it becomes evident that the WAOA-ANN model excels in exploring potential relationships between input and output variables while effectively mitigating overfitting and underfitting issues. Given that the performance of the three hybrid multi-output models is relatively similar, the comparison of their distributions based on the scatterplot may not distinctly distinguish between the model performances. To provide a clearer assessment, the model performance is further illustrated using a Taylor diagram. This diagram facilitates a visual comparison of the three evaluation metrics (R\({}^{2}\), RMSE, and standard deviation) of the models in a single image. Figure 10: Scatter plots of actual and predicted values of \(\theta\) and \(r_{b}/r\) for training and testing datasets: (**a**) WAOA-ANN \(\theta\) prediction; (**b**) WAOA-ANN \(r_{b}/r\) prediction; (**c**) WAOA-RF \(\theta\) prediction; (**d**) WAOA-RF \(r_{b}/r\) prediction; (**e**) WAOA-XGBoost \(\theta\) prediction; (**f**) WAOA-XGBoost \(r_{b}/r\). In a Taylor diagram, the closer the data points are to the measurement point, the better the performance of the model they represent. Meanwhile, to evaluate the model's capacity to leverage potential relationships between outputs, single-output ANN, RF, and XGBoost models were constructed with identical hyperparameter settings for comparative analysis. Figure 11 presents the Taylor diagram of the prediction performance of different models on the test set. As can be seen from the figure, the WAOA-ANN model is closest to the measure point in both \(\theta\) and \(r_{b}/r\) prediction. Additionally, there is a notable enhancement in the prediction performance compared to the single-output ANN model. For the WAOA-RF and WAOA-XGBoost model, on the one hand the predictive performance of the model on the test set is weaker than the WAOA-ANN model.