# Modeling of Tunneling Total Loads Based on Symbolic Regression Algorithm
## 1 Introduction
[22], based on in situ engineering data, the cutter torque is modeled by statistical analysis of the function relationship among parameters, and the coefficient is identified by multiple linear regression. In Ref. [9], data from a number of TBMs are statistically analyzed, and an empirical model of the relationship between the tunneling loads and the diameter of the cutter head is obtained. In Ref. [23], previous studies on data analysis methods are reviewed, and polynomial exponential regression is used to predict the TBM loads. In this kind of regression modeling, it is difficult to determine the nonlinear relationship among the parameters. This is often determined by analyzing the correlation among different elementary functions, but this may lead to an insufficient description of the physical constraint relationship among the parameters. With the increasing abundance of real-time monitoring data and the rapid development of machine learning technology, machine learning modeling based on engineering data is increasing, laying the foundation for determining the coupling relationship between parameters and building load models that fully reflect the nonlinear mapping relationship. Several researchers have modeled tunneling loads based on machine learning algorithms. In Ref. [24], the long short-term memory algorithm is applied to predict the total thrust and cutter head torque in the steady state using a 30-s data window in the rising stage. In Ref. [25], a dynamic tunneling load forecasting method, based on heterogeneous data and data-driven technology, is proposed. In this paper, the random forest algorithm is used to construct a load prediction model based on integrated heterogeneous in situ data. In Ref. [26], based on in situ data from a subway project, a tunneling total load model is established based on particle swarm optimization and a support vector machine. In the above work, various intelligent algorithms are used to model the tunneling load parameters, allowing the influence of multiple factors on the target quantity, and the nonlinear coupling relationship among the influencing factors to be considered. This approach gives good prediction accuracy. The above works reflect the applicability and potential of data-driven technology in the design and analysis of complex engineering systems. However, the current load modeling based on machine learning using engineering data is still basically an \"end-to-end\" black box training method [3; 27], and explicit model functions with some degree of interpretability have not yet been obtained [28]. If an explicit model could be obtained that reasonably describes the causal relationship between parameters under the premise of certain prediction accuracy, it would help to improve the practicability of the prediction model and provide a reference for construction excavation. To solve the above problems, this paper proposes a modeling method for determining the tunneling total loads based on a symbolic regression algorithm. The method combines dimensional analysis and nonlinear feature selection for the symbol regression modeling. First, the physical constraint relationship between the parameters is analyzed from the perspective of dimensions, and the dimensionless parameters (called \(\pi\) quantities) are obtained. Feature selection and symbolic regression modeling is then performed based on these \(\pi\) quantities, which enables the potential physical relationship to constrain the data analysis and modeling process. These processes lay the foundation for obtaining the load models satisfying certain physical rules. Considering the nonlinear coupling relationship between the parameters, this paper selects the input parameters based on the random forest model. Finally, the characteristic \(\pi\) quantities are input to the symbolic regression algorithm for model training. Thus, explicit interpretable load models that satisfy the basic equilibrium theory of mechanics and the dimensional rules of physics are obtained.