# Modeling of Tunneling Total Loads Based on Symbolic Regression Algorithm
## 5 Conclusions  
Aiming at the problem of modeling tunneling loads, a modeling method based on symbolic regression has been proposed in this paper. In this method, dimensionless processing is carried out on the original parameters through dimensional analysis, allowing the nonlinear combination relationship between the parameters to be constructed and the dimensionless parameters (\(\pi\) quantities) to be obtained in accordance with some potential physical connotations. Taking the \(\pi\) quantities as the input parameters for the subsequent calculations and model training is equivalent to providing physical constraints on the training process, and ensures that the modeling process satisfies the relevant dimensional rules, which helps the symbolic regression training to obtain a more efficient and reasonable load model. In addition, the dimensionless processing of the parameters also achieves the effect of dimensionality reduction, reducing the initial nine parameters to six. To further improve the efficiency of the symbolic regression modeling, feature selection is applied before model training to eliminate unnecessary input parameters. To avoid the constraint of a priori assumptions on the function relationship between \(\pi\) quantities, feature selection is based on a random forest model. Finally, the principle of Ockham's razor is applied to complete the symbolic regression modeling. The results show that the method can obtain explicable load models based on the potential physical relationship between the parameters, while guaranteeing the accuracy of the predictions. The model also satisfies the basic mechanical theoretical relations and physical dimensional rules, and the form is simple. This improves the practicability of machine learning modeling, and provides a reference for construction excavation. The modeling method described in this paper provides a new concept for obtaining training models that are interpretable and conform to the relevant physical rules.  
It is important to emphasize that the modeling method combining dimensional analysis, feature selection, and symbolic regression algorithm in this paper can be applied to other projects. Moreover, the models in Equation (9) based on the engineering case of Jilin Project can provide a reference for similar working conditions if needed.  
Conceptualization, Q.Z.; methodology, Q.Z.; funding acquisition, Q.Z.; supervision, Q.Z.; validation, L.Z., S.Z. and S.L.; formal analysis, L.Z.; writing--original draft, L.Z.; writing--review and editing, Q.Z. and L.Z. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Total Thrust**} & \multicolumn{3}{c}{**Total Torque**} \\ \cline{2-9}  & \(\mathbf{R^{2}}\) & **MAPE** & **Model** & \
\begin{tabular}{c} **Theoretical** \\ **Relationship** \\ \end{tabular} & \(\mathbf{R^{2}}\) & **MAPE** & **Model** &
\begin{tabular}{c} **Theoretical** \\ **Relationship** \\ \end{tabular} \\ \hline SR & 0.95 & 1.82\% & Explicit & reasonable & 0.81 & 8.83\% & explicit & reasonable \\ LR & 0.96 & 1.55\% & Explicit & unreasonable & 0.84 & 8.13\% & explicit & unreasonable \\ ANN & 0.95 & 1.71\% & hidden layer & unknown & 0.81 & 8.33\% & hidden layer & unknown \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of tunneling total load models under different modeling methods.  
**Funding:** This research was supported by National Key R&D Program of China [No. 2018 YFB1702505], National Natural Science Foundation of China [No. 11872269], and Natural Science Foundation of Tianjin [No. 18]CYBJC19600].  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.  
**Conflicts of Interest:** The authors declare that they have no conflict of interest to report regarding the present study.