## 5 Conclusions  
This study elaborates on the enhancement strategies for addressing the deficiencies of the Grey Wolf Algorithm, showcasing the successful integration of the refined algorithm with a BP neural network for the inverse analysis of surrounding rock parameters during tunnel excavation processes. The research unfolds across two pivotal sections:  
12. This research thoroughly analyzes the limitations of the Grey Wolf Algorithm and proposes an enhanced version, the Improved Grey Wolf Optimization (IGWO). This upgraded algorithm leverages cubic chaotic mapping and a refraction reverse learning strategy for optimizing the initial population, significantly improving data diversity and uniqueness. Moreover, by substituting the traditional linear convergence factor with a nonlinear one and introducing new parameters such as step size ratio weight and leading wolf weight, the algorithm notably enhances its ability to balance local and global search capabilities. This enhancement not only allows for a more effective escape from local optima but also substantially improves optimization efficiency and accuracy.  
13. In applying this to surrounding rock parameter inversion, the study utilizes an IGWO-optimized BP neural network model, incorporating orthogonal analysis methods to devise 25 numerical simulation experiments. These experiments, which take vault settlement and peripheral convergence data as inputs and output the mechanical parameters of surrounding rock, establish a reverse analysis model. The IGWO-BP neural network model's effectiveness and accuracy in inverting surrounding rock parameters are confirmed by inverse analysis of actual data from YK37 + 330, achieving relative errors of 5.02% in vault settlement and 4.15% in peripheral convergence. These results highlight the model's precision and practicality.  
While this study demonstrates the successful integration of the Improved Grey Wolf Optimization (IGWO) algorithm with the Backpropagation (BP) neural network in the inversion of surrounding rock mechanical parameters, the experimental scenario is relatively simplified, primarily focusing on elastic material behavior. Consequently, the applicability of this method to non-elastic responses remains unverified. Future research will focus on extending the IGWO-BP method to more complex geological and mechanical conditions, with a particular emphasis on non-elastic scenarios, to evaluate its potential for broader applications in rock engineering.  
[PERSON] led the academic direction. [PERSON] developed the manuscript and wrote this paper. [PERSON] conducted the investigation. [PERSON] reviewed this paper. All authors have read and agreed to the published version of the manuscript.  
The authors gratefully acknowledge the financial support from The Science and Technology Program of Hunan Provincial Department of Transportation (no. 202308).  
**Institutional Review Board Statement:** Not applicable.  
Data Availability Statement: The datasets presented in this article are not readily available due to technical limitations. Requests to access the datasets should be directed to [PERSON].  
The authors declare no conflicts of interest.