## 3 Data Description and Evaluation Metrics  
Existing research on borehole breakout can be categorized into numerical simulation and physical experiments in terms of experimental methods. Numerical simulation offers greater convenience in setting up various external conditions to explore the influence of multiple dimensions, making it favored by many scholars. However, the results of physical experiments exhibit better consistency with field reality. In this paper, data obtained from both methods are considered. It is important to note that only data from FEM simulations were collected, as most Discrete Element Method (DEM) simulations were two-dimensional and did not account for the effect of vertical in situ stress [1; 15; 46]. Through literature research [19; 20; 22; 27; 35], the following input features were selected from a wide range of features associated with borehole breakout: rock internal friction angle (\(\varphi\)), cohesion (\(c\)), rock modulus (\(E\)), Poisson's ratio (\(\
u\)), mud pressure in the borehole (\(P_{m}\)), minimum horizontal principal stress (\(\sigma_{h}\)), vertical principal stress (\(\sigma_{v}\)), maximum horizontal principal stress (\(\sigma_{H}\)), and the radius of the borehole (\(r\)). Meanwhile, a total of 501 sets of relevant data were collected from the existing research literature to construct a borehole breakout size dataset [16; 28; 29; 30; 37; 47; 48; 49; 50; 51]. Considering that the absolute magnitude of the borehole breakthrough depth is significantly correlated with the borehole radius, and that the borehole radius setting varies in different studies, the borehole breakthrough depth is dimensionless normalized to improve the model generalization ability. Table 2 summarizes the statistical metrics of the constructed dataset, taking into account the significant variation in borehole sizes involved in different studies, and transforming the borehole breakout depths accordingly. Additionally, it is noted that in the physical experiments, none of them set the mud pressure in the borehole, thus it is considered to be 0 MPa. Figure 5 illustrates the distribution relationship between the breakout sizes. From the figure, it is evident that all the borehole breakout sizes obtained by different researchers exhibit a pronounced nonlinear relationship. This initial observation confirms the feasibility of employing the multi-output approach for borehole breakout size prediction. Additionally, \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Variables**} & \multirow{2}{*}{**Unit**} & \multicolumn{3}{c}{**FEM**} & \multicolumn{3}{c}{**Experiment**} \\ \cline{3-8}  & & **Min** & **Max** & **Mean** & **Min** & **Max** & **Mean** \\ \hline \(\theta\) & \(\circ\) & 0 & 81 & 45.82 & 25 & 138 & 68.04 \\ \hline \(\frac{\eta}{\tau}\) & & 0 & 2.19 & 0.49 & 0.02 & 1.53 & 0.52 \\ \hline \(r\) & cm & 10 & 15 & 13.61 & 0.8 & 1.5 & 1.1 \\ \hline \(\sigma_{H}\) & MPa & 50 & 200 & 103.19 & 35.4 & 93.2 & 61.74 \\ \hline \(\sigma_{v}\) & MPa & 30 & 80 & 47.83 & 5 & 50 & 31.67 \\ \hline \(\sigma_{h}\) & MPa & 20 & 85 & 48.61 & 10 & 40 & 23.54 \\ \hline \(P_{w}\) & MPa & 0 & 45 & 19.03 & 0 & 0 & 0 \\ \hline \(\
u\) & & 0.2 & 0.25 & 0.21 & 0.17 & 0.27 & 0.22 \\ \hline \(E\) & GPa & 14.4 & 59 & 26.83 & 6.66 & 27 & 15.12 \\ \hline \(c\) & MPa & 6 & 55 & 18.66 & 8.92 & 15.5 & 11.83 \\ \hline \(\varphi\) & \(\circ\) & 32.5 & 52.5 & 39.13 & 18 & 39.7 & 31.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical description of features in the datasets. The correlation analysis of the dataset aims to assess whether there is excessive correlation among the input features, potentially leading to multicollinearity issues that could impact the model's performance. As illustrated in Figure 6a, the highest correlation observed between features is 0.86. Notably, for identical input features, the borehole breakout angle and depth exhibit completely opposite correlations.