## 4 Verification Example
Starting from the fourth lamination angle pairs to the 16 th (the last one), one case (\([+45/-45]\)) is considered for even lamination pairs, and two cases (\([0/0]\) and \([90/90]\)) are considered for odd lamination angle pairs. The second, the third, and the fourth pattern sets--\(\mathcal{P}_{2}\), \(\mathcal{P}_{3}\), and \(\mathcal{P}_{4}\)--consist of 16 randomly generated angles of neighboring laminate pairs (ANN inputs) and one corresponding fundamental frequency (ANN output). The number of patterns in \(\mathcal{P}_{2}\), \(\mathcal{P}_{3}\), and \(\mathcal{P}_{4}\) equal 1500, 3500, and 5000, respectively. The pattern sets were applied in different combinations to train the deep network (see Table 2), and the trained network substituted the FE simulations in the GA optimization of the lamination angles. The results reported herein from the GA+ANN optimization procedure, together with the results presented in [41], are shown in Table 2.  
The results presented in Table 2 prove that the proposed GA+ANN optimization procedure is capable of solving the optimization problem. Moreover, even when applying a significantly smaller number of time-consuming FE calls, the results obtained by applying the GA+ANN procedure are still better than those presented in [41]. The proposed procedure is able to solve the optimization problem involving half the number of FE calls, which directly leads to considerable time-savings. Analysis of the results presented in Table 1 leads to the conclusion that there are multiple local maxima \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stacking** & \(f_{1}\) **[Hz]** & \(f_{1}\) **[Hz]** & **Difference** \\
**Sequence** & **Koide and Luersen** & **MODEL3** & **[Hz]** & **[\%]** \\ \hline \([0_{16}]_{s}\) & 1803.4 & 1803.2 & 0.2 & 0.01 \\ \([\pm 45_{16}]_{s}\) & 3450.3 & 3470.8 & 20.5 & 0.59 \\ \([90_{16}]_{s}\) & 2061.2 & 2062.4 & 1.2 & 0.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fundamental frequency calculated by [PERSON] and by using MODEL3. \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **Pattern Sets in** & \(f_{1}\) **[Hz]** & \(f_{1}\) **[Hz]** & **\(\Lambda^{*}\)** & **No. of** \\  & **ANN Training** & **Koide and Luersen** & **MODEL3** & **FE** \\  & & **[PERSON]** [41] & & **Calls** \\ \hline [PERSON]\&[PERSON] & — & 3529.4 & 3522.5 & \([90_{2}/\pm 45_{11}/0/90/\pm 45]_{s}\) & 11,250 \\ GA+ANN & \(\mathcal{P}_{1}+\mathcal{P}_{4}\) & — & 3581.4 & \([90/\pm 45_{13}/0_{2}]_{s}\) & 10,234 \\ GA+ANN & \(\mathcal{P}_{2}+\mathcal{P}_{3}+\mathcal{P  
with \(f_{1}\) equal to or higher than the value found in [41]. The value of \(f_{1}=3581.4\) Hz seems to be a global extreme. Special attention should be given to the results in the rows labeled \"GA+ANN (CLI)\", which were obtained after one curriculum learning loop (see Figure 3). The CL approach was tested with all considered pattern set combinations and, in all of them, gave more precise ANN predictions of \(f_{1}\) fundamental frequency. The cases shown in the rows \"GA+ANN (CLI)\" are those in which the CL approach also improved the optimization result. The most important observation is that the maximum of the fundamental frequency, which is higher than the one presented in [41], was obtained with half the number of FE calls (5100 instead of 11,250).