# Innovative Data-Driven Machine Learning Approaches for Predicting Sandstone True Triaxial Strength
## 6 Performance Comparison
### Comparison on 3D Failure Envelope  
Figure 12 presents the shapes of the failure envelopes and the variation in the \(\frac{\sigma_{1}}{I_{1}}\), \(\frac{\sigma_{2}}{I_{1}}\), and \(\frac{\sigma_{3}}{I_{1}}\) on the deviatoric plane under the parameter conditions of No. 10 sandstone for the six models. The data points in the graph represent actual data points of No. 10 sandstone, with red indicating points within the envelope surface, implying overestimation of rock strength by the model, and green indicating points outside the envelope surface, indicating underestimation of rock strength by the model. From Figure 12a, it can be observed that for the HHO-MLP model, only three data points are in the red state, and they occur at lower stress states. This suggests that the HHO-MLP model predicts the strength of No. 10 sandstone conservatively. Moreover, the shape of the failure envelope is continuous, but with slight roughness and some lack of smoothness at the CMB and TMB. Among the failure envelopes of the six models, the DP and ML models exhibit good performance in terms of smoothness and continuity. However, in terms of differences from the actual data points, the DP model's predictions for actual data in small \(\theta\) ranges are conservative, while  
Figure 10: Performance of six strength models in the meridian plane.  
Figure 12: Cont.  
From Figure 12b, it can be observed that the deviatoric plane shape of the failure envelope of the HHO-MLP model gradually transitions from a triangle to a hexagon, which coincides with the Tresca strength criterion [56]. Meanwhile, at the same \(\theta\) angle, the \(\frac{\sigma_{1}}{I_{1}}\), \(\frac{\sigma_{2}}{I_{1}}\), and \(\frac{\sigma_{3}}{I_{1}}\) vary with the range of \(\theta\) and the trend of \(I_{1}\). Near \(\theta=0^{\circ}\), the \(\frac{\sigma_{1}}{I_{1}}\) decreases continuously as \(I_{1}\) increases. Near \(\theta=60^{\circ}\), the \(\frac{\sigma_{1}}{I_{1}}\), \(\frac{\sigma_{2}}{I_{1}}\) and \(\frac{\sigma_{3}}{I_{1}}\) gradually tend toward a constant value as \(I_{1}\) increases. On the other hand, for the remaining five models, at the same \(\theta\), as \(I_{1}\) increases, the ratios \(\frac{\sigma_{1}}{I_{1}}\), \(\frac{\sigma_{2}}{I_{1}}\), and \(\frac{\sigma_{3}}{I_{1}}\) tend toward a constant value. Thus, it can be observed that for the HHO-MLP model, the growth trend of the triaxial strength in the compression region (\(\theta=0^{\circ}\)) decreases as \(I_{1}\) increases, while the growth trend of the triaxial strength in the tension region (\(\theta=60^{\circ}\)) tends toward a constant value as \(I_{1}\) increases. For the five strength criterion models, the growth trend of the triaxial strength within the same \(\theta\) range becomes constant with the increase in \(I_{1}\), and the shape of the model failure envelope on the deviatoric plane is relatively fixed, only expanding proportionally with the increase in \(I_{1}\).  
However, it is worth noting that, at certain \(I_{1}\) levels, the failure envelope of the HHO-MLP model is not smooth at the CMB and TMB. Of course, only the ML and DP models satisfy this condition across all \(I_{1}\) levels, while the other models exhibit this issue. Overall, from the perspective of the relationship between the failure envelope and the actual data points, the HHO-MLP model demonstrates significant advantages, with decent performance in terms of failure envelope continuity and smoothness. In the entire principal stress space, the HHO-MLP model successfully predicted the true triaxial strength of No. 10 sandstone. However, there are sharp points on the CMB and TMB of the envelope surface at certain \(I_{1}\) levels, leading to discontinuities in the derivatives at these locations,  
Figure 12: Envelope and ternary phases for the six models.  
which could hinder its further application in numerical analysis. This can be solved by mathematical or numerical processing.