## 3 Analyses and Results
### Finite Element Part: Parametric Analysis about Airflow Resistivity and Periodicity Dimension
The three sets share 20 samples, which describe the boundaries of the input space (to avoid extrapolation), to which other examples are randomly added from the aforementioned pool. The remaining elements are used for testing the generalization capabilities of the method. Before proceeding with the learning, feature scaling is executed on all the training samples as:  
\[x_{i}^{\prime}=\frac{x_{i}-\mu}{\sigma_{std}}, \tag{15}\]  
where \(x_{i}\) and \(x_{i}^{\prime}\) are the original i-th feature and the scaled one, respectively, \(\mu\) is the mean value of the feature, and \(\sigma_{std}\) is the standard deviation of the feature, both evaluated on the whole training set. This operation is almost mandatory, since the different orders of magnitude of the parameters involved may generate numerical problems during learning. Feature scaling, instead, reduces all the features to the same order of magnitude. After learning, the features are then descaled, so that the results of the predictions are made understandable to the user. The results provided by Gaussian processes are summarized in Figure 5. Each row displays, on the left, a 3D plot relating the three inputs (airflow resistivity, unit cell dimensions, and resonance frequency), on the cartesian axis, and the predicted peak decrease (color-coded) corresponding to each coordinate. On the right, 2D plots link the independent variables \((\sigma,d)\) (which provide frequency according to Equation (4)) to the corresponding prediction error. In these figures, the black dots indicate the training examples, that must not be considered for the performance evaluation of the machine learning method. Concerning these performances, the 2D plots show that, with a training set made of 45 samples, the prediction error remains inside the 5% boundary. The distribution of the higher errors does not follow a predictable pattern, since they not only appear in the central region of the map, but are also close to the boundaries and, sometimes, training examples. However, the same considerations hold for the lower errors. In order to have greater insight on the performances of Gaussian processes, three test cases are considered: The analytical and numerical TL curves are provided in Figure 6 (where the resonance frequency axis is cut at 14,000 Hz, in order to better visualize the curves and the related peaks), while Table 3 summarizes the results in terms of reference amplitude, predicted amplitude, and prediction error for all the three training sets. Table 3 highlights the excellent performances of Gaussian processes. More generally, TS1, TS2, and TS3 exhibit a mean percent error equal to 1.49, 1.42, and 1.66, respectively, which is much lower than the 5% boundary fixed. Figure 5: On the left, from top to bottom, 3D plots that relate the three inputs of the machine learning algorithm to the predicted TL amplitude percentage increase at the resonance peaks (coded in the colorbar), for each of the three tested TSs. On the right, from top to bottom, 2D plots that relate the two mutually independent inputs of the machine learning algorithm to the percentage error about the prediction of TL amplitude percentage increase at the resonance peaks (coded in the colorbar), for each of the three tested TSs; values marked with black points are those used as training set, while all the remaining ones constitute the actual test set. Machine Learning Part: Prediction of the Resonance Frequency and Transmission Loss Increase at the Resonance Frequency (JCA Model)  
The second characterization task is performed with a meta-material whose foam phase is described by the JCA model. This time, there is no analytical phase, only numerical simulations are executed in order to generate training and test sets. The input features are unit cell dimension, porosity, airflow resistivity, tortuosity, viscous characteristic length, and thermal characteristic length; the output features are TL peak frequency and percent increase. Section 3.4 reports the results of Gaussian processes applied to meta-materials