## 3 Analyses and Results
### Finite Element Part: Parametric Analysis about Airflow Resistivity and Periodicity Dimension
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Test Sample 1** & **Test Sample 2** & **Test Sample 3** \\ \hline
**Reference** & 2724 & 2523 & 1719 \\ \hline
**Prediction 1** & 2724 & 2523 & 1719 \\ \hline
**Percent error 1** & \(1.71\times 10^{-4}\)\% & \(7.04\times 10^{-4}\)\% & 0.0017\% \\ \hline
**Prediction 2** & 2724 & 2523 & 1718.9 \\ \hline
**Percent error 2** & 2.93 \(\times 10^{-4}\)\% & \(7.17\times 10^{-4}\)\% & 0.0048\% \\ \hline
**Prediction 3** & 2723.9 & 2522.9 & 1718.9 \\ \hline
**Percent error 3** & 0.0043\% & 0.0051\% & 0.0054\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Predictions of resonance frequency (in Hz) with five training examples. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Test Sample 1** & **Test Sample 2** & **Test Sample 3** \\ \hline
**Reference** & 45\% & 157\% & 26\% \\ \hline
**Prediction 1** & 45\% & 156.98\% & 26\% \\ \hline
**Percent error 1** & 0.0032\% & 0.0118\% & 0.0030\% \\ \hline
**Prediction 2** & 45.12\% & 157.03\% & 26.08\% \\ \hline
**Percent error 2** & 0.2749\% & 0.0177\% & 0.3226\% \\ \hline
**Prediction 3** & 44.96\% & 156.91\% & 26.01\% \\ \hline
**Percent error 3** & 0.0870\% & 0.0605\% & 0.0421\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Predictions of TL peak percent increase with five training examples. The last test on the performances of Gaussian processes is performed by using five training examples and removing the porosity as feature from the training set, since its values do not change significantly and, more generally, it is proven that it does not affect very much the acoustic behavior of meta-materials described with the JCA model. The predictions of the usual three test samples are listed in Tables 9 and 10, and the error histograms are displayed in Figure 8.  
The results, still excellent, highlight that removing porosity does not affect significantly the performances of Gaussian processes. In particular, focusing on the histograms, the distribution of errors is mainly concentrated in the interval 0-4 \(\times\) 10\({}^{-3}\)% for resonance frequency, and still in the interval 0-0.08% for TL percent peak increase. These outcomes demonstrate that removing a feature, which has negligible effects in the analytical model, \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Test Sample 1** & **Test Sample 2** & **Test Sample 3** \\ \hline
**Reference** & 45\% & 157\% & 26\% \\ \hline
**Prediction 1** & 45.01\% & 157.01\% & 26.01\% \\ \hline
**Percent error 1** & 0.0187\% & 0.0091\% & 0.0209\% \\ \hline
**Prediction 2** & 45\% & 156.98\% & 26\% \\ \hline
**Percent error 2** & 0.0016\% & 0.0143\% & 0.0019\% \\ \hline
**Prediction 3** & 45\% & 156.98\% & 26\% \\ \hline
**Percent error 3** & 0.0017\% & 0.0116\% & 0.0051\% \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Predictions of TL peak percent increase with five training examples and without porosity. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Test Sample 1** & **Test Sample 2** & **Test Sample 3** \\ \hline
**Reference** & 2724 & 2523 & 1719 \\ \hline
**Prediction 1** & 2724 & 2523 & 1719 \\ \hline
**Percent error 1** & 0.0012\% & 0.0016\% & 0.0016\% \\ \hline
**Prediction 2** & 2724 & 2523 & 1718.9 \\ \hline
**Percent error 2** & 0.0013\% & 6.26 \(\times\) 10\({}^{-4}\)% & 0.0081\% \\ \hline
**Prediction 3** & 2723.9 & 2522.9 & 1718.8 \\ \hline
**Percent error 3** & 0.0039\% & 0.0031\% & 0.0119\% \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Predictions of resonance frequency (in Hz) with five training examples and without porosity. Figure 7: Histograms of the percent prediction errors occurrences when predicting the resonance frequency (**a**) and the TL percent peak increase (**b**) with five training examples; these distributions are evaluated on the third training. Each bin contains three occurrences. has, actually, a corresponding outcome in the performances of machine learning algorithms.