## 3 Results and Discussion
### Training Results by ANN Structure  
A MATLAB R2018a toolbox was used to train the artificial neural network. At the beginning of training, initial weights, and biases are generated randomly, so different training results can be obtained for the same structure [59]. In this study, a total of 20 training sessions were performed with each structure, and the average training results are shown in Table 5 and Figure 9. The mean squared error (MSE) was used as the training performance function; it is the one most commonly used in MFNN [60]. The mean squared error (_MSE_) is given by  
\[MSE=\frac{1}{N}\sum_{i=1}^{N}{(T_{i}-K_{i})^{2}} \tag{7}\] \begin{table}
\begin{tabular}{l l} \hline \hline Number of input nodes (strain values) & 55 \\ Number of hidden layers & 1 or 2 \\ Number of hidden nodes & 28, 55, 110, or 111 \\ Number of output nodes (displacement values) & 55 \\ Number of training data & 57,600 \\ Number of test data & 2400 (800 + 800 + 800) \\ Training algorithm & Scaled conjugate gradient backpropagation \\ Transfer function & Logsig (hidden), Purelin (output) \\ Learning rate & 0.01 \\ Momentum & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: General information for the artificial neural network (ANN) model. In Equation (7), \(T\) is the target output and \(K\) is the predicted output. The closer the MSE is to zero, the smaller the error. As the MSE is affected by the absolute magnitude of the output (displacements), there is a limit to judging training performance by MSE only. Therefore, use of the correlation coefficient (\(R\)) value is suggested, and the correlation between the predicted value and the actual value is presented. The closer \(R\) is to 1, the higher the prediction accuracy. The training times for each ANN structure are presented in Table 5. From this table, two features can be found. First, the prediction accuracy varies according to the number of hidden layers and the number of nodes in the hidden layer(s). When there was one hidden layer, the minimum and maximum MSE values were 0.001103 (case 4) and 0.001308 (case 1), respectively. The minimum and maximum \(R\) values were 0.998728 (case 1) and 0.998928 (case 4), respectively. When there were two hidden layers, the minimum and maximum MSE values were 0.001282 (case 7) and 0.001825 (case 5), respectively. The minimum and maximum \(R\) values were 0.998225 (case 5) and 0.998753 (case 7), respectively. From Figure 9, it can be seen that the effect of the ANN architecture on MSE is similar to that on the \(R\) value. In general, the greater the number of hidden layers and nodes, the higher the prediction accuracy. However, as the number continues to increase, the accuracy may decrease because of overtraining. In this study, the learning accuracy was higher in the cases of one hidden layer than in the cases of two hidden layers. It is also seen that for a given number of hidden layers, the greater the number of nodes, the higher the prediction accuracy. The second feature found in Table 5 is that the training time varies according to the number of hidden layers and the number of nodes. For cases 1, 2, 3, and 4, the training time was 840.30 s, 1566.92 s, 3403.87 s, and 3440.74 s, respectively. In case 1, the number of hidden nodes was 28, and in case 2, the number of hidden nodes was 55, which is about twice that for case 1, and the training time was also about twice that for case 1. Similarly, the number of hidden nodes for case 3 was twice as large as that for case 2, and the learning time was also about twice that for case 2. For cases 3 and 4, the number of hidden nodes was similar, so there was no notable difference in training time. For cases 5, 6, 7, and 8, the training time was 997.67 s, 3544.24 s, 16,852.07 s, and 15,442.72 s, respectively. In case 5, the number of hidden-layer nodes was 28, and in case 6, the number of hidden-layer nodes was 55, which is about twice that for case 5, and the number of hidden layers was 2. As a result, the training times differed by a factor of about 3.5.