# Innovative Data-Driven Machine Learning Approaches for Predicting Sandstone True Triaxial Strength
## 1 Introduction  
The strength of rocks, considered one of their fundamental mechanical properties, has been extensively investigated by researchers [1; 2; 3; 4; 5]. In underground engineering construction, rocks experience a complex true triaxial stress state affected by factors like excavation, original rock stress, and support methods. Consequently, developing a robust true triaxial strength model is paramount for accurately evaluating the true triaxial strength of rocks, which is essential for designing and constructing underground engineering projects. Moreover, the efficacy of numerical simulations in geotechnical engineering hinges significantly on the performance of strength constitutive models. To achieve this objective, researchers have proposed a range of theoretical and empirical strength criteria by integrating theoretical analyses and laboratory tests, including uniaxial, biaxial, conventional triaxial, and true triaxial strength assessments. Examples of such criteria encompass the Mohr-Coulomb criterion [6], the Hoek-Brown (HB) criterion [7], and the Bieniawski-Yudbir criterion [8], among others. Notably, the Hoek-Brown criterion and the Mohr-Coulomb criterion are extensively utilized in geotechnical engineering analysis and serve as the foundation for rock material constitutive models in numerousnumerical simulation software packages. However, ongoing experimental advancements have revealed that these strength criteria may not be universally applicable across all rock types, stress states, and failure modes. In particular, the intermediate principal stress significantly influences the true triaxial strength of rocks [9; 10]. Consequently, researchers have focused on developing true triaxial strength criteria that account for the impact of the intermediate principal stress. This involves either refining existing strength criteria to comprehensively incorporate the influence factors of the intermediate principal stress or developing entirely new criteria tailored to this consideration. For example, [PERSON] et al. [11] proposed an enhanced Hoek-Brown criterion by integrating the multifunctional offset function with the original Hoek-Brown criterion meridian function. Similarly, [PERSON] et al. [12] refined both the Mohr-Coulomb criterion and the Hoek-Brown criterion to incorporate the effects of intermediate principal stress. [PERSON] et al. [13] introduced a novel three-dimensional rock strength criterion derived from the Hoek-Brown criterion and the Matsuoka-Nakai criterion. Other strength criteria that take into account the effect of intermediate principal stresses are the Drucker-Prager (DP) criterion [14], the Mogi criterion [15], the Lade-Duncan criterion [16], the modified Weibols-Cook (MWC) criterion [17], the modified Lade (ML) criterion [18], the 3D version of the Hoek-Brown criterion [19], the modified Mohr-Coulomb criterion [20], the Mogi-Coulomb (MGC) criterion [21], etc. Advancements in the true triaxial strength criteria area hold great significance for enhancing the accuracy and reliability of underground engineering design and construction. Theoretical derivation and conventional data fitting have provided valuable insights into true triaxial damage mechanisms in rock. However, exploration in this area remains limited, leaving much potential for improvement in existing strength criteria. In recent years, spurred by the rapid advancements in computer technology, a plethora of machine learning (ML) models have emerged, offering novel avenues for addressing complex nonlinear problems. In the realm of geotechnical engineering, these ML techniques have found extensive application and garnered notable success in analyzing and predicting intricate engineering issues [22]. Moreover, the prediction of rock strength has become a focal point for numerous scholars, who have increasingly turned to ML methods due to their efficacy and versatility.