# NSGA-III-XGBoost-Based Stochastic Reliability Analysis of Deep Soft Rock Tunnel
## 3 NSGA-III Algorithm
Compute the niche count of reference point \(j\in Z^{r}\): \(\rho_{j}=\sum_{\mathbf{s}\in S_{t}/F_{l}}((\pi(\mathbf{s})=j)\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\bm

## 4 Nsga-iii-XgBoost-Based Reliability Calculation Program Framework

### Performance Function of Tunnel Initial Support Structure

According to the design requirements of the perimeter deformation of a deep-buried tunnel, the performance function (limit state function) of tunnel initial support structure \(G_{\text{Tunnel}}(\mathbf{X}^{\prime})\) is given by Equation (5). \[G_{\text{Tunnel}}(\mathbf{X}^{\prime})=U_{\text{max}}-U(\mathbf{X}^{\prime}) \tag{5}\]

where \(U_{\text{max}}\) is the deformation design value of the tunnel initial support structure; \(U(\mathbf{X}^{\prime})\) is the value of the perimeter deformation corresponding to \(\mathbf{X}^{\prime};\mathbf{X}^{\prime}\) is the vector of random variables, that is, surrounding rock geo-mechanical parameters. When designing the excavation section of tunnel, in addition to meeting its clearance and structural dimensions, an appropriate amount of tunnel deformation should be reserved after initial support. The size of the reserved deformation can be determined by the surrounding rock classification, section size, embedded depth, construction method and support conditions, etc., and adjusted sometimes according to the results of on-site monitoring. According to the Chinese Highway Tunnel Design Code, the reserved deformation of the tunnel is generally 5-20 cm. When designing according to the bearing capacity, the allowable relative convergence and the allowable vault settlement for the tunnel initial support structure should be determined by the surrounding rock geological conditions, surrounding rock classification and tunnel buried depth. According to the Chinese Highway Tunnel Design Code, it is generally 0.1-3.0% of the maximum clearance size of tunnel. ### XGBoost-Based Relationship between Geo-Mechanic Parameters and Displacement

In this paper, the XGBoost model was used to surrogate the numerical model and to map the nonlinear relationship between geo-mechanic parameters (elastic modulus, internal friction angle, cohesion and Poisson's ratio, etc.) and monitored displacements, which can significantly reduce the calculation time. The mathematical model \(XGBoost(\mathbf{X}^{\prime})\), is defined by Equation (6). \[\left\{ \begin{array}{l}XGBoost(\mathbf{X}^{\prime})\cdot\mathbf{R}^{K}\to\mathbf{R}^{N}\\ \mathbf{Y}=XGBoost(\mathbf{X}^{\prime})\end{array} \right. \tag{6}\]

where \(\mathbf{X}^{\prime}=(\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{K})\); \(\mathbf{x}_{i}\)\((i=1,2,\cdots,K)\) is a vector of mechanical parameters of surrounding rock such as internal friction angle, elastic modulus, cohesion and Poisson's ratio, etc. ; \(K\) is the dimension of geo-mechanical parameters; \(\mathbf{Y}^{\prime}=(\mathbf{y}_{1},\mathbf{y}_{2},\cdots,\mathbf{y}_{N})\) is an \(N\) dimensional vector of the displacement. In this paper, the observable output is the displacement; correspondingly, \(N\) denotes the dimension of displacement. To obtain \(XGBoost(\mathbf{X}^{\prime})\), a training process based on a dataset is very necessary. The training samples are obtained for this study by combining test design and finite difference numerical simulation, which is used to obtain tunnel displacements according to a given set of surrounding rock mechanical parameters. Compared with other sampling design methods, the Latin hypercube sampling (LHS) method is always comprehensive in considering the marginal small-probability events in the test. Therefore, LHS is more suitable for large-scale numerical test sampling. For this study, the sampling by the LHS sampling method is used to construct a sample set for the geo-mechanical parameters of surrounding rock, and these geo-mechanical parameters are defined as the input of XGBoost. The displacement is defined as the output of XGBoost.