# Modeling of Tunneling Total Loads Based on Symbolic Regression Algorithm
## 4 Case Study
### Discussion  
Analyze the load model Formula (9) and convert it into the form shown in Formula (10). It is found that the thrust and torque models have quadratic and cubic power function relations, with the diameter of the cutter head \(D\), respectively, which conforms to the basic theory of mechanical equilibrium. In addition, the parametric relations given by the models indicate that the ratios of cohesion \(c\), compressive strength \(\sigma_{c}\), and tensile strength \(\sigma_{t}\) to shear modulus \(G\) are the important factors affecting the total loads under similar hard rock geological conditions. The rock cohesion \(c\) and compressive strength \(\sigma_{t}\) have key influences on the total thrust and the total torque, respectively. In the torque model, \(v/\omega\) is actually the penetration degree of the cutter head. Therefore, the penetration degree of the cutter head is also an important factor affecting the total tunneling torque. \[\left\{ \begin{array}{l}F=\left(\frac{c}{G}+\frac{\sigma_{t}}{G}+\frac{\sigma _{c}}{G}\cdot 10^{-2}\right)\cdot 10^{-2}\cdot cD^{2}\\ T=\sigma_{t}\cdot\left(\frac{\sigma}{\omega}D^{2}+10^{-6}\cdot\frac{\sigma_{c }}{G}\cdot D^{3}\right)\end{array} \right. \tag{10}\]  
To further evaluate the tunneling load prediction model, general linear regression and artificial neural network algorithms are used to model and predict the tunneling total loads based on the same engineering data samples. The input parameters are the same as for the symbolic regression. The \(R^{2}\) and MAPE (mean absolute percentage error) of five random samples in the independent test set are selected and averaged. The results are presented in Table 4, and the linear regression model is described by Equations (11) and (12). Analysis shows that the prediction effect of the linear regression model is slightly better than that of the other two algorithms. However, according to Equations (11) and (12), the relationship between certain parameters in the model and the target quantity is obviously inconsistent with the theoretical laws. For example, in Equation (11) there is a negative correlation between the compressive strength \(\sigma_{c}\) of the rock and the total thrust \(F\), and in Equation (12) there is a negative correlation between the cutter head speed \(\omega\) and the total torque \(T\). There is no significant difference in the prediction effect between the symbolic regression model and the neural network model. However, as can be seen from Equation (10) and its analysis, the advantage of the symbolic regression model is that it provides an explicit physical relationship model, which not only conforms to the basic theoretical relationship,  
Figure 10: Statistical results of prediction error on the total set: (**a**) the total thrust, (**b**) the total torque. but also satisfies the physical rules of dimensional uniformity. In conclusion, based on the modeling method proposed in this paper, load models with both prediction accuracy and interpretability can be obtained through the training of engineering data. \[ \begin{array}{l}\frac{F}{GD^{2}}=2.67\cdot 10^{-2}\frac{v}{D\omega}-23.9\mu+9.56 \frac{c}{G}+27.4\omega^{2}D^{2}-0.193\frac{c}{G}+1.77\frac{c}{G}+4.33\\ \Rightarrow F=2.67\cdot 10^{-2}\frac{v}{\omega}GD-23.9\mu GD^{2}+9.56 cD^{2}+27.4\rho \omega^{2}D^{4}-0.193\sigma_{c}D^{2}+1.77\sigma_{t}D^{2}+4.33 GD^{2}\\ \frac{T}{CD^{3}}=1.28\frac{v}{D\omega}+0.257\mu-0.152\frac{c}{G}-0.102\frac{ \omega^{2}D^{2}}{G}+3.67\cdot 10^{-4}\frac{c}{G}+2.53\cdot 10^{-2}\frac{c}{G}-4.85 \cdot 10^{-2}\\ \Rightarrow T=1.28\frac{v}{\omega}GD^{2}+0.257\mu GD^{3}-0.152 cD^{3}-0.102\rho \omega^{2}D^{5}+3.67\cdot 10^{-4}\sigma_{c}D^{3}+2.53\cdot 10^{-2}\sigma_{t}D^{3}-4.85 \cdot 10^{-2}GD^{3}\end{array}  \tag{11}\]  
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