## 4 NSGA-II -XGBoost Based Multi-Objective Back Analysis
### XGBoost-NSGA-II-Based Back Analysis  
We propose the XGBoost-NSGA-II-based back analysis method for the deep soft rock tunnel engineering. The monitored displacements at different monitoring points are denoted herein as \(Y_{\text{monitor1}},Y_{\text{monitor2}},\cdots,Y_{\text{monitord}}\); the predicted displacement by the XGBoost-based surrogate model at different monitoring points are denoted as \(y_{1},y_{2},\cdots,y_{d}\). The non-dominated sorting genetic algorithm-II (NSGA-II) is a fast non-dominated multi-objective optimization algorithm based on the Pareto optimal solution with an elite retention strategy [31; 32]. The objective function of multi-objective optimization is a set of functions containing multiple members, and its optimization form is denoted by Equation (16). \[\mathbf{\theta}^{*}=\underset{\mathbf{\theta}}{\text{argmin}}L(\mathbf{\theta})=\underset{ \mathbf{\theta}}{\text{argmin}}\{\ell_{1}(\mathbf{\theta}),\ell_{2}(\mathbf{\theta}), \cdots,\ell_{d}(\mathbf{\theta})\} \tag{16}\]  
where \(\mathbf{\theta}\) is the geomechanical parameters to be inverted; \"argmin\" represents the variable \(\mathbf{\theta}\) when the objective function \(L(\mathbf{\theta})\) is minimized, i.e., \(\mathbf{\theta}^{*}\); \(d\) is the number of monitoring points for each section of tunnel. An objective function is set for each monitoring point, and each objective function is represented by the mean square error (MSE). Finally, the multi-objective function is denoted by Equation (17). \[\text{min}L(\mathbf{\theta})=\left\{ \begin{array}{l}\text{min}\ell_{1}=(Y_{\text{ monitor}1}-y_{1})^{2}\\ \text{min}\ell_{2}=(Y_{\text{monitor2}}-y_{2})^{2}\\ \text{min}\ell_{d}=(Y_{\text{monitord}}-y_{d})^{2}\end{array} \right. \tag{17}\]  
where \(\ell_{i}\)\((i=1,2,\cdots,d)\) represents the MSE of the \(i\)-th monitoring point. NSGA-II is used to solve the model parameters according to the following steps:  
_Step 1_ Structural representation of solutions. The solution of the model optimization problem may be expressed by a floating point vector or binary vector. The binary vectors are used as chromosomes to represent the true value of the solution space, and the length of the vector is determined by the accuracy of solving the problem. The chromosome number of the initial population is _pop_size_. _Step 2_ Generation of the initial population of feasible solutions. If there is no experience and prior knowledge, the initial population can be randomly generated. _Step 3_ The size of the initial population _pop_size_. The number of chromosomes _pop_size_ is 50-100. By random sampling for _pop_size_ times, the initial sample of the population at \(t=0\) may be obtained. _Step 4_ Evaluation function. Supposing the chromosomes are \(V_{1},V_{2},\cdots,V_{pop\_size}\), an order-based evaluation function is defined by Equation (18). \[\text{eval}(V_{i})=\alpha(1-\alpha)^{i-1}(i=1,2,\cdots,pop\_size,\alpha=0.5) \tag{18}\]  
The _pop_size_ chromosomes are decoded, and the adaptability of each chromosome is evaluated. If the optimal solution remains unchanged for no less than three consecutive generations, the solution is the optimal solution, therefore proceed to Step 9 to obtain the optimal solution. Otherwise, continue to Step 5 to select chromosomes. _Step 5_ Chromosome selection. The selection process of chromosomes is based on spinning the wheel _pop_size_ times. Based on the fitness of each chromosome, each rotation selects a chromosome for the new population. The selection process is as follows:  
First, for each chromosome \(V_{i}\), calculate the cumulative probability \(q_{i}\) by Equation (19). \[\left\{ \begin{array}{l}q_{0}=0\\ q_{i}=\sum\limits_{j=1}^{i}\text{eval}(V_{j})(i=1,2,\cdots,pop\_size)\end{array} \right. \tag{19}\]  
Second, generate a random number \(r\) from \(\left[0,q_{pop\_size}\right]\). If \(q_{i-1}<r\leq q_{i}\), select the \(i\)-th chromosome \(V_{i}(1\leq i\leq pop\_size)\). Third, repeat the above two steps for the _pop_size_ times to get _pop_size_ duplicated chromosomes.