## 5 Informational Based Modeling
### Training the ANN and Methodology  
To train the artificial neural network, a total number of 77 specimens from [36] and [37] was considered in this study, in order to determine two outputs of \(S_{j,ni}\) and \((M_{n}/M_{p,beam})\). Up to \(\approx\) 80% of the selected samples (62) was used for training, while 20% (15 specimens) was used for testing the network. Accordingly, several variables were introduced as input parameters for the model, including:  
* the moment inertia ratio of the column to the connected beam \((I_{col}/I_{b})\);
* the thickness of the top \((th_{tc})\) and bottom \((th_{bc})\) flange cleat;
* the maximum thickness of right or left web cleat \((Max\)-\(th_{uc})\);
* the bolt size \((d_{b})\);
* the ratio of column to beam yield strength \((f_{y,c}/f_{y,b})\). The statistical characteristics of these variables are summarized in Table 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Statistical Index** & **Type** & **Max** & **Min** & **Avg.** & **STD** \\ \hline \(I_{col}/I_{b}\) & Input & 20.00 & 0.29 & 2.59 & 3.95 \\ \(th_{tc}\) & (mm) & Input & 15.90 & 0.00 & 8.10 & 4.79 \\ \(th_{bc}\) & (mm) & Input & 15.90 & 0.00 & 8.70 & 4.45 \\ \(Max\)-\(th_{uc}\) & (mm) & Input & 15.00 & 0.00 & 6.17 & 4.50 \\ \(d_{b}\) & (mm) & Input & 24.00 & 16.00 & 19.51 & 1.69 \\ \(f_{yc}\)\(df_{y,b}\) & Input & 1.13 & 0.79 & 0.99 & 0.08 \\ \hline \(S_{j,ni}\) & (kNm/rad) & Output & 36,365.00 & 1633.00 & 12,021.75 & 9108.08 \\ \(M_{n}/M_{p,beam}\) & Output & 0.95 & 0.13 & 0.43 & 0.19 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of input and output parameters for artificial neural network (ANN) training. The number of hidden layers and the total number of neurons in the hidden layers in an ANN depend on the nature of the problem [51]. Generally, the trial and error method is used to obtain the ideal architecture that best reflects the characteristics of laboratory data. In this paper, an innovative method for calculating the number of neurons in hidden layers is taken into account, namely:  
\[N_{H}\leq 2N+1 \tag{9}\]  
where \(N_{H}\) represents the number of neurons in the hidden layers and \(N_{I}\) is the number of input variables. Since the number of influential input variables for the current study is equal to 6, the empirical Equation (9) shows that the number of neurons in hidden layers can be less than 13. Therefore, several networks with different topologies (with a maximum of two hidden layers and a maximum of 13 neurons) were trained and studied in this paper. The hyperbolic tangent stimulation function and Levenberg-Marquardt training algorithm were used in all networks. The statistical indices used to evaluate the performance of different topologies are the root mean squared error (\(RMSE\)), the average absolute error (\(AAE\%\)), the model efficiency (\(EF\)), and the variance account factor (\(VAF\%\)) that are defined as follows [52]:  
\[RMSE=\left[\frac{1}{n}\sum_{i=1}^{n}(P_{i}-O_{i})^{2}\right]^{\frac{1}{2}} \tag{10}\]  
\[AAE=\frac{\left|\sum_{i=1}^{n}\frac{(O_{i}-P_{i})}{O_{i}}\right|}{n}\times 100 \tag{11}\]  
\[EF=1-\frac{\sum_{i=1}^{n}(P_{i}-O_{i})^{2}}{\sum_{i=1}^{n}\left(\overline{O}_ {i}-O_{i}\right)^{2}} \tag{12}\]  
\[VAF=\left[1-\frac{var(O_{i}-P_{i})}{var(O_{i})}\right]\times 100 \tag{13}\]  
After examining different topologies of networks, it was found that the network with a 6-7-6-1 topology is characterized by the lowest value of error in \(RMSE\), \(AAE\%\), \(EF\), and \(VAF\%\), and by the highest value of \(R^{2}\) to estimate the two output parameters \(S_{j,ini}\) and (\(M_{n}/M_{p,beam}\)), as also shown in Table 3. It is necessary to mention that the error criteria for training and testing the selected data are calculated in the main range of variables and not in the normal range. In this study, as it is for several structural engineering practice applications, the ABC algorithm is used as a new metaheuristic algorithm to determine the weight optimization of each ANN model.