## 7 Conclusions  
To address the contradiction between obtaining the global optimal solution and the high calculation cost in the displacement back-analysis of underground engineering projects, a novel algorithm is proposed for the optimized back-analysis of geomechanical parameters. The main conclusions are as follows:  
1. In this paper, a new intelligent optimization back-analysis algorithm was proposed, which provides a new tool for the rational determination of geomechanical parameters in underground engineering. The algorithm combines the low computational cost and strong fitting ability of the GPR surrogate model with the strong global optimization ability of the GOA. The test results of various benchmark functions showed that the algorithm can effectively guarantee the accuracy of the solution. The evaluation times of the GOA fitness function are greatly decreased, which improves the efficiency of the algorithm.
2. Because a large number of numerical calculations are required when applying the GOA to the back-analysis of geomechanical parameter optimization, resulting in low computational efficiency, a collaborative optimization back-analysis method called GOA-GPR-FLAC\({}^{\text{3D}}\) was proposed. The results of a case study and an engineering example showed that the proposed method has better performance in efficiency and accuracy, and is feasible for the high-dimensional, highly nonlinear back-analysis of geomechanical parameters in underground engineering.  
In order to facilitate engineering applications, this paper only studies the measured displacement back-analysis method based on the classical Mohr-Coulomb constitutive model. In fact, the displacement back-analysis method is only one of the back-analysis methods, and parameters such as stress and strain can also be used as objects of the back-analysis method. In addition, in underground engineering, the deformation of surrounding rock is actually related to many factors, such as rock mass characteristics, support timing and support methods. Therefore, the acquisition of mechanical parameters of underground engineering rock mass for different back-analysis methods and different conditions should be discussed in further research.  
Conceptualization, J.L. and G.S.; methodology, W.S. and Y.Z.; software, W.S. and Y.Z.; validation, J.L., W.S. and G.S.; investigation, W.S.; data curation, W.S.; writing--original draft preparation, W.S.; writing--review and editing, J.L., W.S. and G.S.; visualization, W.S. and Y.Z.; project administration, J.L.; funding acquisition, G.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China for financial support (Grant Nos. 52169021 and 51869003), and the High-Level Innovation Team and Outstanding Scholar Program of Universities in Guangxi province. (Grant No. 202006).  
Not applicable.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} Monitoring \\ Point \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Measured** \\ **Displacement/mm** \\ \end{tabular} } & \multicolumn{2}{c}{**GOA-FLACS\({}^{\text{3D}}\)**} & \multicolumn{2}{c}{**GOA-GPR-FLAC\({}^{\text{3D}}\)**} \\ \cline{2-6}  & & **Calculated** & **Related** & **Calculated** & **Related** \\  & & **Displacement/mm** & **Error/\%** & **Displacement/mm** & **Error/\%** \\ \hline M7 & 5.833 & 5.377 & 7.81 & 5.749 & 1.44 \\ M19 & 4.044 & 4.680 & 15.72 & 4.151 & 2.65 \\ M24 & 1.633 & 1.645 & 0.73 & 1.632 & 0.06 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of displacements obtained from calculation and measurement.