# NSGA-III-XGBoost-Based Stochastic Reliability Analysis of Deep Soft Rock Tunnel
## 1 Introduction  
After the excavation of deep-buried soft rock tunnels under high stress, the risk of large deformation of soft rock often occurs. Therefore, it is very necessary to quantitatively evaluate the large deformation risk using reliability theory. The reliability-based analysis is more rational to deal with parametric uncertainties and their randomness. Therefore, it has attracted extensive research interest in recent years [1; 2; 3; 4]. Evaluating the reliability of tunnel structures is a key tool for assessing the safety level and their most probable failure state under possible uncertainties during their service life. According to the theory of structural reliability, the safety level of engineering structures can be denoted as a probabilistic model, and its mathematical formula is as follows [5; 6]:  
\[P_{\text{f}}=Prob[g(\mathbf{X})\leq 0]=\int\cdots\int_{g(\mathbf{X})\leq 0}f_{ \text{X}}(x_{1},x_{2},\cdots,x_{n})dx_{1}dx_{2}\cdots dx_{n} \tag{1}\]  
where \(P_{\text{f}}\) denotes the probability of structure failure; _Prob_ represents the probability operator; \(g(\mathbf{X})\) stands for the limit state function of tunnel structure; \(\mathbf{X}\) denotes the vector of random variables; \([g(\mathbf{X})<0]\) indicates tunnel structural failure; \(f_{\text{X}}(x_{1},x_{2},\cdots,x_{n})\) denotes the joint probability density function (PDF) of \(\mathbf{X}\). The traditional reliability analysis methods include the first-order method, the second-order method and the response surface method. The system reliability of rock tunnel wasassessed using an iterative procedure based on the convergence-confinement method, the first-order reliability method and the response surface method [7]. The reliability of the tunnel face was assessed using adaptive RBF and first-order reliability method [8]. The reliability of tunnel face and the tunnel reliability were analyzed by the response surface methods based on least squares support vector machines [9] and high-dimension model representation [10]. In addition, there are other innovative methods [11; 12; 13; 14; 15; 16; 17; 18; 19; 20] used in reliability research of tunnel engineering. The surrogate models based on augmented radial basis functions [11] and a hybrid particle swarm optimization neural network [12] were used to assess the reliability analysis of tunnels and the probability distribution of the limit support pressure for the tunnel excavation face. The reliability of the tunnel face was assessed using the active learning Kriging model [13], the hybrid evolution Markov chain Monte Carlo (MCMC) algorithm [15] and the simplified inverse first-order reliability method (FORM) [20]. The system reliability of the tunnel reinforced by rock bolts was analyzed using the improved hybrid method of the linear response surface method and artificial neural network (ANN) [16]. The reliability of the tunnel was analyzed using mixed polynomial correlation function expansion, adaptive sequential experimental design and an adaptive algorithm [17]. The performance of the tunnel composite lining was analyzed using the improved Roenblueth point estimation method, FORM, Monte Carlo sampling method and finite element method (FEM) [18]. The stability reliability of the tunnel lining structure was analyzed using the optimization method for the Kriging interpolation in collaboration with the genetic algorithm (GA) [19]. With further study of the uncertainty in geotechnical engineering, the reliability method has been gradually developed and applied in geotechnical engineering. However, there are still few studies on tunnel reliability, only relatively concentrated on the above aspects [20]. It is significant to note that the subject of structural reliability analysis in civil engineering has recently been further developed [21; 22; 23]. To incorporate extra information into the Gaussian process as constraints, a Bayesian-entropy Gaussian process methodology for regression and surrogate modeling was proposed [21].