## 1 Introduction
When the DBM model is used, its simplicity allows to perform the analytical design of the meta-material assembly, which returns the unit cell dimension and the airflow resistivity on the basis of target thickness and resonance frequency; successively, the TL peak percent increase, with respect to the homogeneous configuration, is numerically estimated. Both the analytical and numerical procedures are used to generate the training set, after which the characterization of the meta-material is performed via a machine learning approach. Successively, a similar characterization task is performed, in which the porous material is described with the JCA model, that is an acceptable trade-off between description accuracy of the material and model complexity. In more detail, in this task all the model's parameters (namely, unit cell dimension, porosity, airflow resistivity, tortuosity, viscous characteristic length, and thermal characteristic length) are provided as input features for machine learning methods, which predicts the resonance frequency of the TL peak, first, and the percent increase of the TL peak itself, then. The use of two models describing a meta-material has a two-fold purpose: The investigation of machine learning prediction capabilities when a \"simple\" (DBM) and a \"complex\" (JCA) model is used, and the performance changes when the number of features and training samples vary according to the model. Indeed, the results prove that more complex material models allow a finer description in machine learning terms which leads to better results. Gaussian processes are chosen as a machine learning method for this work, mainly due to their suitability to regression problems. In the second place, but not less importantly, Gaussian processes are mathematically equivalent to many well-known models (like Bayesian linear models) and are closely related to others, such as support vector machines. Moreover, the model provided is easier to handle and interpret than other counterparts and, at the same time, the method itself is easier to apply practically (it does not require plenty of decisions like architecture, activation functions, learning rate, etc.) [40; 41]. Therefore, the aim of this work is to investigate the applicability of more manageable, practical, and interpretable machine learning methods, such as Gaussian processes, for the characterization of foam-based meta-materials, demonstrating, at the same time, the improvement of prediction performances, even though more complex phenomenological material modeling is involved. Furthermore, the effectiveness in characterization tasks is also highlighted, which may boost up the preliminary phase of material design, characterization, and optimization, contextually reducing financial expenses and computational time related to experimental tests and numerical simulations. This work is structured as follows. First of all, the theoretical frameworks related to the characterization of the meta-material, by means of DBM and JCA models (Sections 2.1.1 and 2.1.2, respectively), and to Gaussian processes (Section 2.2) are provided. Section 3 is dedicated to the analysis of results, in which data pre-processing and the application of the machine learning method are accomplished differently, according to the meta-material model. In particular, Section 3.1 illustrates the procedure that allows to analytically calculate the required values of foam airflow resistivity and unit cell dimension when the DBM model is considered. After this design part, in Section 3.2, the properties of the studied acoustic package are introduced, along with the 3-dimensional Finite Element (FE) geometries, and a parametric test campaign is performed for several setups, and test articles described with both the foam model. Successively, in Section 3.3, Gaussian processes are applied in order to predict the TL increases at the resonance frequencies when the meta-material is described by the DBM model; the resulting error is smaller than 5%.