# Innovative Data-Driven Machine Learning Approaches for Predicting Sandstone True Triaxial Strength
## 6 Performance Comparison  
### Comparisons Using the Collection Dataset  
In order to evaluate the performance of the obtained model, we compared the HHO-MLP model with five existing rock strength criteria (DP, HB, MGC, ML, and MWC) on both the training and testing datasets, considering that these five strength criteria each involve parameters related to rock properties such as cohesion and friction angle, and that such parameters are not explicitly provided in the literature from which the dataset originated. To ensure a fair comparison of model performance, the parameters of the various strength criteria were fitted using the data from the training set through the least squares method, following the methodology outlined by [PERSON] et al. (2017) and [PERSON] et al. (2015). The parameter settings obtained for the five different rock types through fitting are presented in Table 3.  
The obtained HHO-MLP model and the five strength criteria were used to predict the training set data and test set data. Figure 8 shows the comparison between the actual values and the predicted values obtained from the HHO-MLP model and the five strength criteria. From Figure 8, it can be seen that the scatter plots of the HHO-MLP model are closely surrounding the Y = X line in both the training and test sets, and the upper and right distributions are similar in shape. The scatter plots of the remaining five strength criterion models show large dispersion, especially for \(\sigma_{1}\) > 400 Mpa. This demonstrates the strong generalization ability and good prediction accuracy of HHO-MLP for various sandstone types as well as stress conditions. The traditional strength criteria have inapplicable rock types and stress conditions with poor generalization performance. Furthermore, the figure illustrates that the predicted values remain constant across different intermediate principal stress conditions due to the Hoek-Brown criterion's failure to incorporate the influence of intermediate principal stresses. Figure 7: Performance of HHO-MLP models utilizing different activation functions in the meridian plane: (**a**) sigmoid, (**b**) softplus, (**c**) swish, (**d**) tanh. To quantitatively compare the performance of the six strength models, four commonly used evaluation metrics were selected to assess the models' predictive performance. These evaluation metrics are R\({}^{2}\) (Coefficient of Determination), MAE (Mean Absolute Error), RMSE (Root Mean Square Error), and MAPE (Mean Absolute Percentage Error). The R\({}^{2}\) was used to evaluate the degree of agreement between predicted values and actual values in regression models [51; 52; 53], and it can be calculated using Equation (43). \[R^{2}=\frac{\sum_{i=1}^{n}(\overline{y}-f_{i})^{2}}{\sum_{i=1}^{n}(y  
The MAE accurately reflects the magnitude of errors between predicted values and actual values [54], and it can be calculated using Equation (44).  
\[MAE=\frac{1}{n}\sum_{i=0}^{n}\lvert f_{i}-y_{i}\rvert \tag{44}\]  
RMSE denotes the standard deviation of the fitting error between the predicted and actual values, and it can be calculated using Equation (45). \[RMSE=\sqrt{\frac{\sum_{i=1}^{n}\left(f_{i}-y_{i}\right)^{2}}{n}} \tag{45}\]  
The MAPE represents the proportion of errors between predicted values and actual values relative to the actual values, and it can be calculated using Equation (46). \[MAPE=\frac{1}{n}\sum_{i=0}^{n}\left\lvert\frac{f_{i}-y_{i}}{y_{i}}\right\rvert \tag{46}\]  
Figure 8: Scatter plots of actual and predicted values of \(\sigma_{1}\) for training and testing datasets: (**a**) HHO-MLP model, (**b**) DP criterion, (**c**) HB criterion, (**d**) MGC criterion, (**e**) ML criterion, (**f**) MWC criterion. Table 4 presents an overview of the evaluation metrics for the predictions made by the six strength models. It is evident from the table that the HHO-MLP model marginally underperforms the MGC concerning MAPE in the training set evaluation (HHO-MLP: 0.1435, MGC: 0.1382).