## 4 Conclusions  
The scope of this work is to develop and validate tools for boosting up the characterization of vibroacoustic packages based on foam cores with embedded periodic patterns through the implementation of machine learning algorithms.  
First of all, the theoretical frameworks related to the characterization of the metamaterial and to the chosen machine learning method are provided. Then, a procedure that allows to analytically calculate the required values of foam airflow resistivity and unit cell dimension in order to reach a desired target in terms of periodic resonance frequency is illustrated under specified constraints. Successively, the properties of the studied acoustic package are introduced, together with the 3-dimensional finite element geometries, and a parametric test campaign is performed for several setups, each with different values of foam airflow resistivity and unit cell dimension. In conclusion, a Gaussian-based machine learning algorithm is developed and applied in order to predict, with an error smaller than 5%, the Transmission Loss increases at the resonance frequencies when the DBM model is used to describe the meta-material. When the JCA model is considered, the Gaussian process can provide even better results, in predicting resonance frequencies and TL peak percent increases, with much less training examples, by increasing the number of features characterizing the observations.  
Therefore, machine learning methods that are easy to handle and interpret, such as Gaussian processes, provide several advantages when meta-materials packages need characterization. In fact, the switch from DBM to JCA models demonstrates that the increase of input features leads to a noticeable reduction of the number of training examples. Hence, if one wants to generate an experimental training set, such a reduction translates into fewer samples to generate and test, and thus less money and more time savings. Moreover, important time savings are introduced in numerical simulations, since it would imply less modeling and simulations to run. All these elements help in providing automated and fast estimations of acoustic indicators, which are useful especially for investigating new configurations in preliminary design phases.  
Future expansions of the present work may involve the development and implementation of more complex formulations for the description of materials, as well as advanced machine learning techniques, not only for the estimation of the acoustic performance of meta-materials, but even for their choice and optimization, according to the applications of interest. Last, but not least, it has been shown how increasing the number of features leads to an astounding level of accuracy and, at the same time, an appreciable decrease of the number of training examples. These results open the path towards the creation of an  
Figure 8: Histograms of the percent prediction errors occurrences when predicting the resonance frequency (**a**) and the TL percent peak increase (**b**) with five training examples and without porosity; these distributions are evaluated on the third training. Each bin contains three occurrences.  
experimental training set, since the limited number of training samples would imply fewer experiments, thus time and money saving.  
Conceptualization, A.C. and D.M.; Data curation, A.C. and D.M.; Investigation, A.C. and D.M.; Methodology, A.C. and D.M.; Resources, D.M.; Supervision, A.C., D.M. and G.P.; Validation, A.C.; Visualization, A.C. and D.M.; Writing--original draft, A.C. and D.M.; Writing--review & editing, G.P., F.F. and S.D.R. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Italian Ministry of Education, University and Research (MIUR) through the project DEVISU, under the scheme PRIN-2017, grant agreement No. 22017 ZX9X4K006.  
Not applicable.  
The authors declare no conflict of interest.