Borehole Breakout Prediction Based on Multi-Output Machine Learning Models Using the Walrus Optimization Algorithm  
[PERSON]  
1 School of Resources and Safety Engineering, Central South University, Changsha 410083, China; [EMAIL_ADDRESS]. embarked on establishing the correlation between borehole breakout sizes and in situ stress, aiming to further elucidate the mechanism of borehole breakout. For example, [PERSON] et al. [10] combined the Mohr-Coulomb criterion with the Kirsch equation to develop a correlation model between in situ stress and borehole breakout size. [PERSON] [11] analyzed the vertical borehole stress distribution by linear elastic model and evaluated the rock stress state by Mogi-Coulomb criterion to construct the correlation between borehole mud pressure, borehole breakout and in situ stress. However, because the process of borehole breakout is a continuous development, the stresses in the borehole wall and surrounding area will be further redistributed after the fractured rock is exfoliated, which will result in new rock exfoliation. Therefore, the models proposed by [PERSON] et al. [10] and [PERSON] [11] have certain limitations. The advancement of numerical simulation methods has offered an effective approach to addressing this issue, with several researchers successfully simulating the continuous development process of borehole breakout [12; 13; 14]. In recent years, numerous researchers have also been investigating borehole breakout through numerical simulation methods. For instance, in 2019, [PERSON] et al. [15] examined the influence of borehole size and temperature on borehole breakout development using Discrete Element Method (DEM). They explored the pattern of microcrack development in the borehole wall and its surroundings under varying borehole sizes and temperatures. [PERSON] et al. [16] conducted a study on the impacts of borehole breakouts on breakdown pressure using finite element method (FEM) simulation. They established a correlation between in situ stress, borehole breakout size, and breakdown pressure. [PERSON] et al. [17] employed a three-dimensional bonded-particle model to simulate borehole breakout and successfully replicated the V-shape breakout phenomenon. From the existing experimental, theoretical, and numerical simulation analyses and studies on borehole breakout, it is evident that the size of borehole breakout is influenced by factors such as mineral composition, porosity, grain strength, intergranular strength, angle of internal friction, cohesion, borehole mud pressure, temperature, in situ stress, Poisson's ratio, rock modulus, and borehole size [18; 19; 20; 21; 22]. To achieve accurate assessment of borehole breakout, it is apparent that all relevant features should be considered as comprehensively as possible. The intricate nonlinear relationship among numerous features poses a significant challenge for conventional experimental and theoretical analyses, while numerical simulation is also complex and challenging. The emergence of machine learning (ML) methods offers a novel approach to address this issue. Existing ML methods possess exceptional learning capabilities for capturing complex nonlinear relationships among different factors. Consequently, they have been widely applied by researchers for investigating engineering problems involving multi-feature  
Figure 1: Dog-ear shaped borehole breakout. considerations [23; 24; 25]. Furthermore, in [PERSON] et al. 's study, data from different wells were used to train the prediction model for the physical data of the rock at the drill bit during drilling, and the results had fairly good accuracy. This shows that machine learning models are well able to find potential nonlinear relationships between data from mixed datasets [26]. In the domain of borehole breakout assessment, ML methods have also seen extensive application. Table 1 collects some existing studies on ML for constructing borehole breakout association models, detailing the features, data sources, and algorithms they employed.