# Innovative Data-Driven Machine Learning Approaches for Predicting Sandstone True Triaxial Strength
## 1 Introduction
By leveraging large volumes of rock experimental data for learning and training purposes, ML models can discern intricate patterns underlying rock strength and provide precise predictions. This methodology not only improves the accuracy of rock strength prediction but also mitigates human errors effectively throughout the prediction process. In the study conducted by [PERSON] et al. [23], nine predictive models for rock true triaxial strength were established using six distinct ML algorithms. The research findings suggest that, among these nine models, seven exhibit significant superiority over the existing strength criteria. Furthermore, the enhanced GP model proposed by [PERSON] et al. [24] achieves precise predictions of various rock true triaxial strengths. Similarly, the GEP model developed by [PERSON] et al. [25] showcases robust generalization capabilities in predicting true triaxial strengths across different rock types. Concurrently, artificial neural network (ANN) models have also seen widespread application in the prediction of rock true triaxial strengths [26; 27; 28]. In current research utilizing ML to forecast rock true triaxial strength, several issues have been identified. For instance, the majority of ML models employed by researchers are black-box models, lacking the ability to produce explicit expressions. This limitation hinders the broader extension and application of the derived models. Additionally, some researchers have attempted to address this limitation by creating explicit expressions of the models, leveraging the characteristics of the ML models they utilized. For instance, [PERSON]'s, [PERSON]'s, and [PERSON]'s studies provided explicit expressions of the models. However, due to the limitations of the datasets utilized, wherein stress states predominantly involve compression, [PERSON]'s and [PERSON]'s studies failed to accurately predict the true triaxial strength of rocks under tensile principal stress states (\(\sigma<0\)). This inadequacy clearly falls short of meeting practical application needs. While [PERSON]'s ANN model theoretically enablespredictions across various stress states and provides an explicit expression of the model, their study evaluation is confined to comparing the predictive accuracy of the model on the dataset. Consequently, there is a lack of assessment regarding the model's performance in the principal stress space, making it challenging to ascertain whether the predicted trend of strength development under tensile states aligns with theoretical expectations. For the true triaxial strength model of rocks, it is crucial to satisfy differentiability for further applications in numerical analysis. The Multilayer Perceptron (MLP) model, as a type of feedforward artificial neural network model, is differentiable everywhere except for individual points wherein certain activation functions are used. This provides a foundation for the further expansion of the model's applications. Additionally, the MLP model has shown promising performance in predicting mountain slope failures [29], sand-over-clay bearing capacity [30], and mining-induced stress in underground mines [31]. This study introduces a novel approach by developing a true triaxial strength model for sandstone using a Multilayer Perceptron (MLP) optimized with the Harris hawks optimization (HHO) technique. The MLP model is designed to provide explicit, continuous, and differentiable expressions, which are essential for numerical analysis applications. A dataset comprising 417 sets of true triaxial experimental data for sandstone was collected from the existing literature and randomly divided into a training set (70%) and a testing set (30%). During the model training process, 80% of the training set data were utilized for model training, with the remaining 20% used for validating the model's performance.