# Innovative Data-Driven Machine Learning Approaches for Predicting Sandstone True Triaxial Strength
## 7 Conclusions  
This study proposes a method for constructing a true triaxial strength model for sandstone using an MLP model. The HHO technique is employed to adjust the hyperparameters of the MLP model. By using the \(\sigma_{c}\), \(\sigma_{2}\), and \(\sigma_{3}\) as inputs, the prediction of the true triaxial strength of sandstone is achieved. Crucially, the study ensures that the MLP model is continuously differentiable, meeting the requirements for numerical analysis by carefully adjusting the model's structure and activation function. Four distinct HHO-MLP models were trained using true triaxial strength data of sandstone with different activation functions. Instead of solely relying on prediction accuracy for model selection, an innovative comparison was conducted by analyzing the models on the meridional plane. This analysis excluded two models with strong prediction accuracy but significant discrepancies in the compressive and tensile meridian trends compared to theoretical expectations. Among the remaining models, the one that best aligned with theoretical trends while maintaining high prediction accuracy was chosen as the target HHO-MLP model. Five existing strength criteria were used as a comparison (DP, HB, MGC, ML, MWC). The results indicate that the HHO-MLP model outperforms the five existing strength criterion models on both the training and test sets. HHO-MLP shows better generalization performance for various sandstones under various stress conditions. In contrast, the predictions of \(\sigma_{1}\) for the \(\sigma_{1}>400\) MPa state by the traditional strength criteria all show large deviations, which indicates that the generalization performance of the traditional strength criteria is weak. At the same time, the large deviation in the traditional strength criteria in the high-stress zone suggests a change in the true triaxial strength behavior of the sandstone with increasing stress. The conventional strength criteria are limited by their structure and cannot predict this change process. Across the four evaluation metrics used, the HHO-MLP model slightly lags behind the MGC model in terms of MAPE on both the training and test sets, but outperforms the five strength criterion models in terms of R\({}^{2}\), RMAE, and MAE. This demonstrates the accuracy of the HHO-MLP model in predicting the true triaxial strength of sandstones at the stress states involved in the dataset. With further analysis of the shape of the HHO-MLP model's envelope in the meridional plane, deviatoric plane, and principal stress space, it was observed that the compressive and tensile meridians of the HHO-MLP model align well with experimental data, and its development trend is similar to that of the HB, MGC, ML, and MWC models, except for minor differences in the compressive meridian under high \(S_{8}\) conditions. On the deviatoric plane, the shape of the HHO-MLP model's failure envelope matches different strength criteria at various stress levels and generally meets the requirements of smoothness and continuity. A comparison in the principal stress space reveals that the HHO-MLP model predicts rock true triaxial strength conservatively, further validating the continuity and smoothness of the failure envelope. However, in the compressive region (\(\theta=0^{\circ}\)), the predicted development trend of rock true triaxial strength by the HHO-MLP model differs from that of the other five strength criterion models as \(I_{1}\) increases. Nonetheless, due to the lack of true triaxial test data for this rock in this region under high \(I_{1}\), it is challenging to determine which trend better fits the actual situation. In summary, this study demonstrates that the obtained HHO-MLP model outperforms the five existing strength criterion models in terms of prediction accuracy. Meanwhile, the HHO-MLP model has a better performance on the failure envelope and completely predicts the true triaxial strength of sandstone No. 10 in the principal stress space.