## 5 Conclusions  
Borehole breakout, as a critical factor influencing borehole quality, has consistently attracted significant attention from researchers. This study collected a total of 501 datasets from numerical simulations and physical experiments, encompassing in situ stress, borehole dimensions, rock properties of the borehole wall, and mud pressure in the borehole as input features, with borehole breakout size as the output data. Three hybrid multi-output models (WAOA-ANN, WAOA-RF, WAOA-XGBoost) were developed using ANN, RF, and XGBoost as base algorithms, augmented by the WAOA optimization technique. The predictive performance of these models was compared and analyzed, while a single-output model was constructed to further assess the efficacy of the multi-output framework. The principal findings of this investigation are delineated below:  
(1) All three hybrid multi-output models exhibit excellent prediction performance for borehole breakout size. Particularly, the WAOA-ANN model demonstrates consistent performance across both the training and test sets, surpassing WAOA-RF and WAOA-XGBoost on the test set (for \(\theta\) training (test): R\({}^{2}\) = 0.9906 (0.9819), MAE = 0.0095 (0.0142), VAF = 0.9803 (0.9629), RMSE = 0.018 (0.0244); for \(r_{b}/r\) training (test): R\({}^{2}\) = 0.9839 (0.978), MAE = 0.0167 (0.0208), VAF = 0.9679 (0.9556), RMSE = 0.0284 (0.0334)). Consequently, the WAOA-ANN model is deemed to possess superior prediction capabilities and stronger generalization ability. (2) The WAOA-ANN model effectively leverages the potential correlation between borehole breakout sizes, resulting in a substantial enhancement in prediction performance compared to the single-output model. Conversely, WAOA-RF and WAOA-XGBoost exhibited no improvement in performance relative to the single-output model with identical hyperparameter settings, while also increasing the computational resource requirements of the model. (3) The sensitivity analyses conducted on WAOA-ANN indicate that for both \(\theta\) and \(r_{b}/r\), \(\sigma_{H}\) exhibits the highest sensitivity. However, there exists a considerable disparity in the rankings of input features concerning the impact on \(\theta\) and \(r_{b}/r\) (\(\theta\): \(\sigma_{H}>c>p_{w}>\
u>E>r>\varphi>\sigma_{h}>\sigma_{v}\), \(r_{b}/r\): \(\sigma_{H}>p_{w}>c>\sigma_{v}>E>\varphi>r>\
u>\sigma_{h}\)). Additionally, variations are observed in the correlation of the same feature with \(\theta\) and \(r_{b}/r\). For instance, \(\sigma_{v}\) displays a positive correlation with \(\theta\) but a negative correlation with \(r_{b}/r\). Similarly, \(\
u\) and \(\sigma_{h}\) exhibit a positive correlation with \(\theta\) but a nonmonotonically correlation with \(r_{b}/r\). In conclusion, all three hybrid multi-output models developed achieved effective prediction of borehole breakout size, but only the WAOA-ANN model successfully demonstrated the advantages of the multi-output structure and performed better in terms of overall prediction performance. Therefore, the obtained WAOA-ANN is considered to be an effective hybrid multi-output model for borehole breakout size prediction. It is important to note that prediction results may be biased if the input feature values used exceed the range of the dataset in this paper. Hence, subsequent studies may consider covering a more comprehensive range of eigenvalues to enhance the robustness of the predictions. R.Z., methodology, validation, resources, visualization, software, and writing--original draft; J.Z., conceptualization, methodology, validation, investigation, visualization, writing--review and editing, supervision, and funding acquisition; M.T., formal analysis and writing--review and editing; C.L., investigation and writing--review and editing; P.L., writing--review and editing; T.L., validation, formal analysis, writing--review and editing, and supervision. All authors have read and agreed to the published version of the manuscript.