*3.2. Inferring Effective Properties*

We prepared 5 samples, four of them were used in the training stage, represented in Figure 3, in which the pores volume fraction was kept constant (*φ* = 0.5) and the spatial distribution almost uniform.

The constructed nonlinear regression (based on the use of *Code2Vect*) described in the previous section, is now applied to the sample shown in Figure 4 where while keeping the same almost uniform pore distribution and the same volume fraction, hexagons and heptagons were randomly mixed. In this same figure, the solution of the thermal problem at the microscopic scale for obtaining the effective thermal conductivity that will serve as reference value, is also included. Finally, it also shows both the PD and the PI.

The PI, **<sup>y</sup>** <sup>∈</sup> <sup>R</sup>*D*, is then projected into the three retained orthonormal PCA modes to give the thee weights that constitute the data *<sup>ξ</sup>* <sup>∈</sup> <sup>R</sup><sup>3</sup> (*d* = 3) to be processed by the nonlinear regression (based on the *Code2Vect*) that produces vector **<sup>z</sup>** <sup>∈</sup> <sup>R</sup><sup>2</sup> (we enforce a 2D representation, *q* = 2, for the sake of clarity in the data visualization)

$$\mathbf{z} = \mathbf{W}(\mathfrak{F}) \mathfrak{F} \tag{23}$$

and then identify the set S(**z**) of data **z***<sup>i</sup>* closest to **z**, from which the QoI, the effective thermal conductivity, is interpolated

$$\mathcal{O} = \sum\_{i \in \mathcal{S}(\mathbf{z})} \mathcal{F}(\mathbf{z}, \mathbf{z}\_i) \, \mathcal{O}\_{i\nu} \tag{24}$$

with in the present case O ≡ *K*<sup>22</sup> and with radial bases as interpolation functions F(**z**, **z***i*).

Figure 5 places **z** with respect to its neighbors, where color scales with the target quantity, that is, with *K*22. The inferred value of the effective thermal conductivity *K*<sup>22</sup> using Equation (24) for the micorstructure depcited in Figure 4 results *K*22(**z**) = 73.4 W/mK, very close to the reference value computed numerically from the temperature distribution shown also in Figure 4, of *K*22,REF = 74 W/mK.

**Figure 4.** (**a.1**) Histogram of the pores radius; (**a.2**) considered microstructure; (**a.3**) temperature field used for computing the effective thermal conductivity that will serve as reference for evaluating the regression performance; (**a.4**) persistence diagram; and (**a.5**) persistence image.

**Figure 5.** Interpolation space **z** with color scaling with the values of the effective thermal conductivity *K*22.
