## 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  & **Case 1** & **Case 2** & **Case 3** \\ \hline \((\sigma,d)\) & (13,925, 0.022) & (8348, 0.0174) & (23,228, 0.0137) \\ \hline
**Resonance frequency** & & & \\
[Hz] & 6680 & 8900 & 10,515 \\ \hline
**Amplitude increment** & & & \\
**(FEM) [\%]** & 42.38 & 78.09 & 42.17 \\ \hline
**Amplitude increment** & & & \\
**(ML-TS1) [\%]** & 42.32 & 79.22 & 41.61 \\ \hline
**Error on amplitude** & & & \\
**increment (ML-TS1) [\%]** & 0.13 & 1.44 & 1.33 \\ \hline
**Amplitude increment** & & & \\
**(ML-TS2) [\%]** & 42.66 & 79.42 & 41.72 \\ \hline
**Error on amplitude** & & & \\
**increment (ML-TS2) [\%]** & 0.66 & 1.70 & 1.06 \\ \hline
**Amplitude increment** & & & \\
**(ML-TS3) [\%]** & 42.69 & 79.40 & 41.31 \\ \hline
**Error on amplitude** & & & \\
**increment (ML-TS3) [\%]** & 0.72 & 1.67 & 2.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Amplitude increments at the resonance peaks and related errors for three tested (\(\sigma\), \(d\)) couples, predicted with machine learning through three different training sets (ML-TS). Figure 6: Transmission Loss analytically (homogeneous) and numerically (meta-core) computed, respectively through TMM and FE approaches, for three test cases with different combinations of airflow resistivity and unit cell dimension; further details about Case 1, Case 2, and Case 3 are reported in Table 3. The TL percent increase is estimated on meta-core curves, using the homogeneous curves as reference. described with JCA models. In this Section, Gaussian processes are used to predict the resonance frequency and TL percent increase (at the predicted resonance frequency) of meta-materials described with the JCA model. The JCA model allows to describe a meta-material in terms of six variables, namely: Unit cell dimension \(d\), porosity \(\phi\), airflow resistivity \(\sigma\), tortuosity \(\alpha_{\infty}\), viscous characteristic length \(\lambda_{v}\), and thermal characteristic length \(\lambda_{th}\). From a machine learning point of view, such a plethora of parameters is useful fo reducing the number of training samples, thus it is expected that good predictions can be obtained with a limited number of examples for the training phase. The number of training examples is 38. Due to the high dimensionality of feature space, the description of  
The results highlight the high accuracy of the predictions the Gaussian processes are capable of providing. Comparing these outcomes with those obtained with the DBM model, since both the characterization of the Gaussian process and the number of training samples are the same (or, more precisely, the number of training samples is almost the same, being even smaller for the JCA case), it is clear that the main cause behind this improvement of performances is due to the finer characterization allowed by the JCA model, providing six input features against three in the case of the DBM model. Hence, it is reasonable to continue the investigation in order to understand the change of the performances of the machine learning algorithm when reducing the number of training examples. Quite surprisingly, it is possible to reduce the training set to 5 samples (less samples would bring to numerical errors in the procedure) and still obtain the same quality of results, as highlighted in Tables 7 and 8, reporting the predictions and errors of the same test samples. Also in this case, the predictions accuracy is very high, exhibiting a very small percent error. Moreover, a larger picture concerning the distribution of errors (in the case of the third training) in the remaining 30 test samples is provided by the histograms in Figure 7.  
The histograms show that, in general, the errors are very low for all the other test samples. Particularly, the errors in predicting the resonance frequency are mainly concentrated in the interval 0-4\(\times 10^{-3}\)%, while those obtained when predicting the TL peak increase are almost totally in the interval 0-0.08%.