# Reliability and Efficiency of Metamodel for Numerical Back Analysis of Tunnel Excavation
## 5 Results and Discussions  
### Metamodel and Prediction Results for Test Datasets  
The structure of the metamodel of each group is expressed in Table 4. Metamodels developed from A to E groups are named Metamodels A, B, C, D, and E, respectively. Loss values were expected to decrease as the number of values increased but the loss value of Metamodel A was lower than those of Metamodels B and C. This situation can be considered a type of noise called overfitting. Overfitting means that the ANNs model has not been generalized since it is so accurately tailored to the training and verification datasets. Figure 4 shows the minimum, maximum, and average losses and mean absolute percent error (MAPE) of a metamodel of each group. In Figure 4, min and max imply the lowest and highest loss values in 400 models developed for grid search of each group. Average implies the average loss value of the 400 models of each group. These three indexes are based on the left Y-axis. MAPE is the evaluation index expressed by Equation (5). The MAPE represented by yellow diamonds is based on the right Y-axis.  
\[\text{MAPE}=\frac{1}{n}\sum_{i=1}^{n}\left|\frac{\delta_{ti}-\delta_{pi}}{ \delta_{ti}}\right| \tag{5}\]  
Except for the minimum value, as we expected, the loss of the average and maximum value decreased as the number of values per property increased. These tendencies imply that Metamodel A is a special case of predicting validation datasets well. The prediction degree of the metamodels was evaluated using test datasets of each group by MAPE. Every metamodel shows a MAPE value of less than 2%. This result could imply that the metamodel created using three values per parameter is also reliable for representing FLAC. However, all of the parameter values in the test datasets exist in the training datasets. It is possible that these metamodels are good at predicting only for datasets with property values existing in training datasets. That is, overfitting must be checked. Therefore, further verifications are required using new datasets that utilize other property values.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Metamodel**} & \multicolumn{2}{c}{**Number of Nodes**} & \multicolumn{2}{c}{**Activation Function**} & \multirow{2}{*}{
\begin{tabular}{c} **Loss** \\ **(m)** \\ \end{tabular} } & \multirow{2}{*}{**MAPE (\%)**} \\ \cline{2-2} \cline{5-6}  & **1 st Hidden Layer** & & & & & \\ \hline A & 20 & 8 & S & R & 0.007341 & 1.46 \\ B & 8 & 20 & L & R & 0.01434 & 1.84 \\ C & 8 & 11 & R & H & 0.01395 & 1.27 \\ D & 20 & 20 & H & S & 0.00552 & 0.50 \\ E & 25 & 20 & H & R & 0.00329 & 0.37 \\ \hline \hline \end{tabular}  
H: hyperbolic tangent; L: Leakyrelu; R: ReLU; S: sigmoid.  
\end{table}
Table 4: Structures of the metamodels and test results.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Equation** & **Reference** \\ \hline \(3\text{Ni}_{i}/2\) & [PERSON] [23] \\ \(2\text{Ni}_{i}+1\) & [PERSON] [24] \\ \(3\text{Ni}_{i}\) & [PERSON] [25] \\ \(4\text{Ni}_{i}\), \(5\text{Ni}_{i}\) & [PERSON] and [PERSON] [26] \\ \hline \hline \end{tabular}
\end{table}
Table 3: Equations for determining the numbers of nodes.