## 5 Back-Analysis of Rock Parameters with GOA-GPR-FLAC\({}^{\text{3D}}\)  
### Proposed Method  
The optimization back-analysis of displacement finds the best combination of parameters by iterative optimization and makes the calculated displacement value approach the measured displacement value step-by-step while correcting the parameters to be determined. The objective function of the optimization back-analysis of displacement can be set in the following form:  
\[f(\mathbf{X})=\sqrt{\sum_{i=1}^{n}\left[d_{i}(\mathbf{X})-d_{i}^{*}\right]^{2}} \tag{13}\]  
where \(\mathbf{X}\) is a set of parameters to be inverted; \(d_{i}(\mathbf{X})\) is the calculated displacement value of the first monitoring point on the rock and soil mass; \(d_{i}^{*}\) is the measured displacement value of the \(i\)-th monitoring point; \(n\) is the number of total displacement monitoring points. The essence of an objective function is the mean square error (MSE) that reflects displacement. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **GOA** & **GOA-GPR** \\ \hline The optimal value/\(10^{-4}\) & 9.18 & 9.73 \\ The worst value/\(10^{-4}\) & 9.51 & 9.94 \\ Average number of function evaluations & 369,600 & 37,612 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of calculation results for Case 2. Figure 6: Three-dimensional space surface diagram of Case 2. Figure 7: Convergence curve of the test function in Case 2. The ultimate goal of the optimization back-analysis of displacement is to find a set of rock mechanical parameters \(\mathbf{X}\) so that the value of the objective function can be minimized, and the parameter \(\mathbf{X}\) corresponding to the minimum value can be considered the \"equivalent mechanical parameter\" of the on-site rock. FLAC\({}^{\text{3D}}\), which was first developed by Itasca Consulting Group, Inc. in Minneapolis, Minnesota, the United States, is a numerical analysis software based on the three-dimensional explicit finite difference method. FLAC\({}^{\text{3D}}\) is widely used in mechanical calculations in the field of geotechnical engineering due to its wide application range, simple embedded language and powerful pre- and post-processing functions. An optimization back-analysis method that combines the GOA-GPR cooperative optimization algorithm with the FLAC\({}^{\text{3D}}\) numerical method is proposed in this section and as GOA-GPR-FLAC\({}^{\text{3D}}\). This method integrates the FLAC\({}^{\text{3D}}\) program into the GOA-GPR cooperative optimization algorithm and provides a new method for the back-analysis of geomechanical parameters. The specific implementation steps of the algorithm in this section are as follows:  
Step 1: Initialize the parameters of the GPR model and determine the value range of the geomechanical parameters. Step 2: Within the parameter value range, the GOA is used to randomly generate \(N\) initialization operators with parameters, and the FLAC\({}^{\text{3D}}\) software is used to calculate the corresponding objective function value of the initial operator to obtain the fitness value of the initial operator. Then, determine whether the grasshopper operator in the historical value set satisfies the convergence criterion. If so, output the corresponding operator information; otherwise, go to Step 3. Step 3: During the GOA global optimization process, all the operator information is recorded in the historical database. Step 4: In the historical dataset, the current optimal operator is selected, and \(3\times N\) optimal operators are selected from the entire historical dataset to construct a global optimization experience dataset. Step 5: GPR is used to model the global optimization experience set and solve the first and second derivative information of the GPR approximate fitness function at the same time. An initial point is randomly selected within the parameter value range of the surrogate model, and the iterative formula of [PERSON]'s method is used for iterative optimization.