*3.3. Microstructures with Varying Shapes and Size Distribution*

These first preliminary successful results were pushed forward by considering quite more complex microstructures. Thus, a total of 35 samples were generated, while varying other parameters, in particular pore size (following uniform and Gamma distributions) and pore shape (circular or 5 to 8 side polygons, randomly chosen). The volume fraction was kept constant (*φ* = 0.5). 34 samples were used in the training, keeping one, the one shown in Figure 6, for inferring the thermal conductivity and concluding on the ability of the proposed technique to infer accurately it. Figure 7 places the considered microstructure in the **z**-space where the thermal conductivity is interpolated, to infer the value of *K*<sup>22</sup> = 81 W/mK, for a reference value of *K*22,REF = 78 W/mK.

**Figure 6.** (**a.1**) Histogram of pores radius; (**a.2**) testing microstructure; (**a.3**) temperature field used for calculating the reference effective thermal conductivity; (**a.4**) persistence diagram; and (**a.5**) persistence image.

**Figure 7.** Interpolation space **z** with color scaling with the values of the effective thermal conductivity *K*22.

To check the prediction improvement with the sampling richness, the effective thermal conductivity in the microstructure shown in Figure 6 while considering different samplings in the training stage, from 13 to 35 microstructures, with the relative errors reported in Table 1.

**Table 1.** Relative error in the effective conductivity prediction depending on the number of samples considered in the regression (training stage).

