## 1 Introduction  
In the design process of lining support structures, it is generally necessary to construct a mechanical analysis model to analyze the deformation and stability of the tunnel in detail [1, 2]. Accurate mechanical parameters of the rock mass are the basis for establishing the mechanical analysis model. Influenced by factors such as stratum lithology, buried depth, geological structure, groundwater, and ground stress, the mechanical parameters of rock mass have a certain degree of randomness and spatial variability [3, 4]. These characteristics are directly reflected in the uncertainty of the mechanical behavior of the surrounding rock, which makes the bearing capacity and deformation prediction of the tunnel structure extremely complex. Traditional tunnel support design theories and safety assessment methods often rely on empirical engineering analogy methods and deterministic verification design methods. Randomness and spatial differences in material parameters are indirectly considered by retaining a certain safety margin [5, 6]. These methods have certain limitations, which cannot guarantee the absolute safety of engineering structures, and it is difficult to quantify various uncertainties. The reliability theory based on probability statistics can effectively solve the uncertainty problem in tunnel construction. Through the reliability analysis method, combined with the mechanical calculation model to output the failure probability or reliability index, the safety and reliability of the engineering structure can be quantitatively evaluated. Thus, subjective bias and overly conservative solutionsin traditional design can be effectively avoided, providing technical guidance for the safe construction of future tunnel projects [7; 8]. At present, the commonly used reliability analysis methods of engineering structures include Monte Carlo Simulation (MCS), the statistical moment method, stochastic finite element (SFEM), etc. [9; 10; 11; 12]. The MCS method is the most widely used, but it has a large number of calculations and low efficiency. The statistical moment method is more efficient than the MCS method, but its calculation accuracy is low and there are some limitations. The difficulty of the SFEM method is determining how to reasonably select random variables in the high-dimensional parameter space and reduce the number of calculations. With the progress and development of technology, the reliability analysis method based on a surrogate model has been widely used because of its advantages of computational efficiency and accuracy. At present, the commonly used surrogate models are the response surface method (RSM), Kriging model, polynomial chaos expansion (PCE) method, artificial neural network (ANN), and support vector machine (SVM) [13; 14; 15; 16; 17]. Compared to other surrogate models, the significant feature of the Kriging model is that it can not only provide estimates for unknown functions but also provide error estimates for the estimated values. Therefore, this model has received widespread attention from researchers and is currently one of the most representative surrogate models [18]. In the Kriging model, improper hyper-parameters of the covariance function may lead to over-fitting or under-fitting of the model. In order to further enhance the overall performance of the Kriging model, many scholars use swarm intelligence algorithms to optimize and improve the hyper-parameters of the model [19; 20]. [PERSON] et al. [21] used the ant colony optimization algorithm to optimize the parameters of the Kriging model and realized the modeling of the main discontinuities of rock tunnels. [PERSON] et al. [22] used PSO, GA, the grey wolf algorithm, and the firefly algorithm to optimize the Kriging surrogate model. It is worth noting that the accuracy of the Kriging model depends, to some extent, on the selection of sample points. For the complex high-pressure tunnel support structure model, a large number of sample points need to be selected, which will result in a large number of calculations.