## 4 Experiments and Analyses  
### Inverse Analysis of Tunnel Surrounding Rock Mechanical Parameters Based on IGWO-BP  
The neural network utilizes a tri-layer architecture for its computations, with the hidden layer's neuron count \(m\) being ascertainable through the following empirical equation:  
\[m=\sqrt{a+n}+l \tag{15}\]  
where \(a\) signifies the neuron count in the input layer, which is influenced by the number of displacement indicators and is thus 2. The variable \(n\), indicating the neuron count in the output layer, depends on the number of inversion parameters, amounting to 4. Lastly, \(l\) stands as a constant valued between 1 and 10. The transition from the input to the hidden layer employs the logsig function--an S-shaped hyperbolic function--whereas the  
Figure 5: Comparison of function plot and convergence curve of multi–peak function. (**a**) \(\mathrm{f_{6}}\)–Schwefel 2.26 Function; (**b**) \(\mathrm{f_{7}}\)–Rastrigin Function; (**c**) \(\mathrm{f_{8}}\)–Ackley Function; (**d**) \(\mathrm{f_{9}}\)–Griewank Function; (**e**) \(\mathrm{f_{10}}\)–Goldstein Price Function.  
connection from the hidden layer to the output layer uses the purelin function--a linear function. Following this formula and through preliminary calculations, the hidden layer's neuron count is finalized at 7.  
The neural network structure model established for the inverse analysis of surrounding rock parameters of the Lotus Tunnel is illustrated in Figure 6 below.  
The results from the 25 experimental configurations, identified via orthogonal testing, have been compiled. These results include both crown settlement and peripheral convergence displacement values, which serve as the input vectors for training. Concurrently, the mechanical parameters from each of these configurations are utilized as output vectors for training. This approach creates the IGWO-BP neural network's training dataset, detailed in Table 3.  
Utilizing the IGWO-BP neural network post-training, it is possible to inversely deduce the mechanical parameters of the surrounding rock \(\left[\text{E},\,\mu,\varphi,\text{c}\right]\) from rock deformation data, including crown settlement and peripheral convergence. This method completes the reverse analysis procedure. To assess the trained neural network's reliability, real-world measurements from the YK37 + 330 section were used for verification. By inputting the measured values of crown settlement and peripheral convergence displacement from YK37 + 330 into the trained inversion model, the model provided predictions of the surrounding rock's mechanical characteristics: an elastic modulus of 0.32 GPa, a Poisson's ratio of 0.26, a cohesion of 156.14 kPa, and an internal friction angle of 28.52\({}^{\circ}\).  
This study adopts a simplified experimental scenario, focusing primarily on the behavior of elastic materials. The selection of this scenario aims to validate the basic effectiveness of the integration of the Improved Grey Wolf Optimization (IGWO) algorithm and the Backpropagation (BP) neural network in the inversion of surrounding rock mechanical parameters. Given the simplicity of the chosen scenario, the applicability of the results is currently limited to elastic material behavior, and further validation for non-elastic responses will require more complex experimental setups.  
Figure 6: Neural network model of inversion of surrounding rock.