## 3 Simulation Models
### Initial Model  
This section explains the way an ANN model was designed and developed in order to predict the seepage rate. To design an effective intelligent system, an important step is the proper use of input data. The literature was reviewed to identify the parameters that can affect the seepage rate. Table 2 presents the input data applied to the proposed model. Every intelligent network consists of two basic parts: training and testing. The former creates a nonlinear relationship between independent and dependent variables, while the latter examines the first part. Therefore, out of the total 24,000 pieces of data accessible here, there is a need to determine a portion applicable to training processes and another to testing processes. As widely suggested in the literature by various researchers, [55; 56; 57; 58; 33; 54; 55; 56; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85; 86; 87; 88; 89; 90; 91; 92; 93; 94; 95; 96; 97; 98; 99; 100; 101; 102; 103; 104]. A variety of algorithms have been introduced in the literature for the purpose of training ANN, among which a key one is the Levenberg-Marquardt (LM) method [100; 101; 102; 103; 104]. It can be effectively applied to problems that may arise in the civil engineering and mining contexts [60; 61; 62]. Three layers exist in this model; the first and third layers are related to the input and output data, respectively. In studies carried out formerly, a hidden layer has been also utilized for solving many linear and nonlinear engineering problems [105; 106; 107; 108]. For that reason, in this study, a hidden layer is also used. Figure 5 presents the ANN structure implemented in this paper. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model No.**} & \multirow{2}{*}{**Structure**} & \multicolumn{2}{c}{**The Best R\({}^{2}\)**} \\ \cline{3-4}  & & **Training** & **Testing** \\ \hline T1 & 6-1-1 & 0.7925 & 0.8050 \\ T2 & 6-2-1 & 0.9126 & 0.9068 \\ T3 & 6-3-1 & 0.8738 & 0.8633 \\ T4 & 6-4-1 & 0.9018 & 0.9184 \\ T5 & 6-5-1 & 0.9378 & 0.9333 \\ T6 & 6-6-1 & 0.9140 & 0.9050 \\ T7 & 6-7-1 & 0.8919 & 0.8998 \\ T8 & 6-8-1 & 0.9071 & 0.9199 \\ T9 & 6-9-1 & 0.9155 & 0.9143 \\ T10 & 6-10-1 & 0.9241 & 0.9077 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of various artificial neural network (ANN) models with activation function Tanh. The principal objective of the ANN model is the exploration of a proper way through which the intelligent systems could perform their defined predictive tasks. Typically, two parameters have impact on performance of the ANN models: the number of neurons and the number of iterations. More specifically, the number of neurons existing in the hidden layer meaningfully contributes to determination of the minimum limits of computational space. This study continues toward the phases that are set on the basis of the initial conditions of the under-consideration problem. If the initial conditions cannot be met, the system then continues working on the basis of a predefined number of iterations. The literature consists of various approaches to determining the optimum number of neurons that can exist in the hidden layer. To this end, different researchers have suggested different formulas [104; 109; 110]. Nevertheless, in the engineering field, numerous studies have implemented various numbers of data, which shows that the intelligent models demonstrate their best performance in every computational space with a certain number of neurons [105; 111]. Moreover, a complete analysis of 10 to 100 iterations was done in order to specify the number of iterations. The value of 80 iterations was selected by sensitivity analysis as the value that requires the best results. Additionally, R\({}^{2}\) and RMSE were employed as statistical indices in order to evaluate the developed model's performance quality.