# Modeling of Tunneling Total Loads Based on Symbolic Regression Algorithm
## 2 Modeling Method Based on Symbolic Regression Algorithm  
The modeling method for the tunneling load proposed in this paper mainly uses the symbolic regression algorithm to obtain explicit models. In addition, the method combines dimensional analysis and nonlinear feature selection into symbolic regression modeling, which can realize total load modeling with interpretability and prediction accuracy by constructing reasonable \(\pi\) quantities, applying suitable feature selection methods, and setting appropriate hyperparameters. Eventually, the model also can satisfy the basic equilibrium relations of mechanics and the dimensional rules of physics. Figure 1 shows the overall block diagram of the proposed method, which consists of two main stages: the construction of \(\pi\) quantities based on dimensional analysis, and the quantitative identification of \(\pi\) quantities based on in situ tunneling data. The quantitative identification of \(\pi\) quantities includes three parts: in situ data preprocessing, feature selection, and model training. First, based on the basic idea of the dimensional analysis \(\Pi\) theorem, the tunneling loads and the parameters affecting them are analyzed, and the corresponding dimensionless quantities (\(\pi\) quantities) are calculated. The \(\pi\) quantities will be used as the input parameters for subsequent analysis and calculations. Then, in the second stage, the in situ tunneling data are preprocessed, and then the feature parameters are selected based on the random forest algorithm. These parameters are used as the input parameters for subsequent model training. In the last step, the symbolic regression algorithm is applied to train the models based on the tunneling data. In essence, the construction of \(\pi\) quantity is a nonlinear combination based on the dimensional relationship between parameters, which can mine the potential physical relationship to a certain extent and satisfy dimensional homogeneity. For any physical relationship, dimensional homogeneity should be maintained. It is often difficult to satisfy dimensional rules and physical laws if the original physical parameters are input to a machine learning algorithm directly to train a model. Therefore, the dimensional analysis of each parameter is carried out before the model training stage. In this process, it is necessary to determine the fundamental quantities of all the parameters, based on which, dimensionless operations are performed on other parameters. The fundamental quantities need to be determined by combining the characteristics of the specific problem. Thus, this paper combines the concept of the dimension itself and the physical meaning of tunneling loads. The advantage of using \(\pi\) quantities as the input parameters in subsequent calculations, compared with directly inputting the influential parameters, is that the machine learning training process is constrained through certain physical relations. It is these physical relationships that lay the foundation for obtaining load models that have a certain physical connotation and satisfy the dimensional rules. As in situ tunneling data have outliers, the measured data are preprocessed in order to mine the common laws that affect the total loads from the tunneling data and avoid the interference of abnormal data in the model training. In the data preprocessing, an outlier identification technique based on the difference method combined with box diagrams is proposed, to retain as much information from the original data as possible. This technique does not make any prior assumptions about the data distribution. First, the difference value of each data point is obtained by calculating the forward and backward difference of the tunneling loads. The outliers of the difference values are then identified by the box diagram method. Finally, the corresponding data points after the union of the difference outliers are determined as abnormal data and removed.