# Inverse Modeling of Seepage Parameters Based on an Improved Gray Wolf Optimizer
## Improved Gray Wolf Optimizer
### Numerical Experiment of Algorithm Performance  
A numerical experiment was performed to verify the effectiveness of IGWO. Six typical functions were selected for simulation in the experiments, including Sphere, Rosenbrock, Quartic, Rastrigin, Ackley, and Griewank. Table 1 shows the mathematical expressions, dimensions, and search ranges of these typical functions. Sphere, Rosenbrock, and Quartic are single-peak functions. Especially Quartic is a multidimensional flat bottom function with random disturbances. The single-peak functions are mainly applied to determine the accuracy of IGWO. Rastrigin, Ackley, and Griewank are multi-peaked functions that tend to cause the algorithm to converge to a locally optimal solution. The performance to jump out of the local optimum could be tested reasonably for IGWO. In addition, the experimental results of IGWO, SGWO [33], and GWO are compared in the simulation. To ensure a fairness, the population size is 30, and the maximum number of iterations is 500 for IGWO, SGWO, and GWO. The three algorithms were performed 30 times independently for each function, and the average values were taken as the simulation results. The results of the numerical experiments are given in Table 2. The optimal values simulated by IGWO are closer to the theoretical optimal solution than the results of the other two algorithms. Among them, the simulation results of Sphere, Rastrigin, and Griewank are equal to the theoretical values, revealing the high accuracy of IGWO. Figure 2 presents the convergence curves of the six test functions with IGWO, SGWO, and GWO. The convergence curves of all three algorithms continue to decrease as the number of iterations increases. In particular, the convergence curve of IGWO decreases significantly faster than the corresponding curves of the other two algorithms, indicating the progress of IGWO in terms of running time. Furthermore, for the multi-peaked test function, the convergence curve of IGWO continues to decrease while the search of the other two algorithms converges to a local optimum solution. This result demonstrates the ability to jump out of the local search and converge to the global optimum value in IGWO. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test** & **Mathematical** & **Dimension** & **Search Range** \\
**Function** & **Expression** & **Dimension** & **Search Range** \\ \hline Sphere & \(\sum\limits_{i=1}^{D}x_{i}^{2}\) & 30 & [–100, 100] \\ Rosenbrock & \(\sum\limits_{i=1}^{D-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+\left(x_{i} -1\right)^{2}\right]\) & 30 & [–30, 30] \\ Quartic & \(\sum\limits_{i=1}^{D}ix_{i}^{4}+random[0,\ 1]\) & 30 & [–1.28, 1.28] \\ Rastrigin & \(\sum\limits_{i=1}^{D-1}\left[x_{i}^{2}-10 cos(2\pi x_{i})+10\right]\) & 30 & [–5.12, 5.12] \\ Ackley & \(-20 exp\left(-0.2\sqrt{\frac{1}{30}\sum\limits_{i=1}^{D}x_{i}^{2}}\right)-exp \left(\frac{1}{30}\sum\limits_{i=1}^{D}cos2\pi x_{i}\right)+20+e\) & 30 & [–32, 32] \\ [PERSON] & \(\frac{1}{4000}\sum\limits_{i=1}^{D}x_{i}^{2}-\prod\limits_{i=1}^{D}cos\left( \frac{x_{i}}{\sqrt{t}}\right)+1\) & 30 & [–600, 600] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test functions in the numerical experiment.