## 3 Verification through a Simple Beam Under Sinusoidal Loads
### Optimization Results  
As proposed in Section 2.2, the location of n\({}_{\text{s}}\) strain-measurement points was optimized in the order of four, three, and two. The GA optimized the location of the strain-measurement points as optimization variables. The MSE of the trained ANN was used as an objective function. The ANN architecture of n\({}_{\text{s}}\)-n\({}_{\text{s}}\)-41 (no. of inputs-no. of hidden nodes-no. of outputs) was selected. The population size was 40, and generations were iterated until the minimum fitness value was not updated over more than 50 iterations. Table 3 shows the optimization results. Means squared error (MSE), and regression value (R) were used as the results of ANN training and testing. MSE is a function that shows the ANN training accuracy in general. The closer to 0, the higher the accuracy. Since MSE may be influenced by the absolute values of the predicted values, the regression (R) value was additionally used to evaluate the training accuracy. The R-value represents the degree of correlation between the target and the prediction. An R-value tending to 1 represents a high correlation. Figure 6 shows the optimized locations for each number of strain-measurement points on the beam. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Period (T)** & **\(\Delta\)t (s)** & **Datasets (= 1 s/\(\Delta\)t)** \\ \hline
1 s (training) & 0.00625 & 161 \\
0.5 s (training) & 0.003125 & 321 \\
0.1 s (training) & 0.000625 & 1601 \\
0.8 s (test) & 0.005 & 201 \\ \hline \multicolumn{2}{c}{**Total number of datasets**} & - & 2083 (training), 201 (test) \\ \hline \multicolumn{2}{c}{One dataset consists of 41 strains (input) and 41 displacements (output).} \\ \end{tabular}
\end{table} 
Table 2: The number of datasets for ANN training and testing. Figure 5: Sine loads applied to the simple beam. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Locations of Strain-Measurement Points**} & \multicolumn{3}{c}{**Training**} & \multicolumn{1}{c}{**Test**} \\ \cline{3-8}
**No. of Strains (= n\({}_{\text{s}}\))** & \multicolumn{3}{c}{**(mm)**} & \multicolumn{3}{c}{**Training**} & \multicolumn{1}{c}{**Test**} \\ \cline{3-8}  & \multicolumn{3}{c}{**Point 1**} & **Point 2** & **Point 3** & **Point 4** & **MSE** & **R** & **MSE** \\ \hline \multirow{3}{*}{**Arbitrary**} & 4 & 75 & 150 & 2775 & 2850 & 8.0591 & 0.9945 & 7.4406 \\  & 3 & 75 & 2850 & 2925 & - & 14.9554 & 0.9895 & 16.3313 \\  & 2 & 150 & 300 & - & - & 16.7763 & 0.9886 & 16.3592 \\ \hline \multirow{3}{*}{**Optimized**} & 4 & 450 & 1500 & 2250 & 2700 & 0.0113 & 0.9999 & 0.0125 \\  & 3 & 900 & 1500 & 2250 & - & 0.0073 & 0.9999 & 0.0072 \\ \cline{1-1}  & 2 & 1125 & 2025 & - & - & 0.0513 & 0.9999 & 0.0524 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The results of ANN training and testing with optimized strain-measurement points. When four strains at the optimized locations (450, 1500, 2250, and 2700 mm) were used as the ANN input, the MSE and R of the trained ANN were 0.0113 and 0.9999, respectively. For comparison, the MSE and R-values are presented in Table 3 when four strains at arbitrary locations (75, 150, 2775, and 2850 mm) were used as the ANN input. The MSE and R of the trained ANN were 8.0591 and 0.9945, respectively. Even if the same number of strains were used, it can be seen that the prediction accuracy would differ significantly depending on the location. Likewise, a similar tendency was shown when three and two strains were used. In the case of the strains at the optimized locations, the MSE was 0.0113, 0.0073, and 0.0513 for four, three, and two strains, respectively. The R-value was 0.9999 in all cases. The MSE close to 0 and the R close to 1 showed that the overall displacement prediction accuracy was very high even when using two strains at the optimized locations. Although the ANN was well-trained, the prediction accuracy of the data that was not used in the training was significantly lower than the training accuracy. This situation is called overfitting.