## 4 Description of the Application Problem and Employed Multi-Fidelity Models
### Surrogate Modeling for Space Design Exploration
#### 5.3.4 Calculation of Error Metrics and Assessment of Surrogate Models
Figure 7 shows the calibration plots for the observed and predicted (metamodeled) values of the intermediate variable \(\Delta\), and the equivalent plot in terms of the target variable \(\dot{m}_{cond\_3D}\). This latter calibration plot has been obtained on the basis of the surrogate of \(\Delta\) and the LF model \(\dot{m}_{cond\_0D}\) by means of Equation (17). Considering the training dataset (green triangles), the surrogate is constructed for the intermediate variable (see Figure 7) with a \(R^{2}(z_{m})\)=0.95, increasing the correlation up to \(R^{2}(z_{t})\)=0.98 when representing the target variable. For the verification dataset (blue circles), the error for the difference is much greater than that of the target variable, in which a few points lie outside the depicted 10% dispersion band. A close-up of Figure 7 shows that the performance of the surrogate is less adequate close to the origin. However, this region corresponds to low mass flow rates, so their contribution to the absolute error is low (notice that the normalized \(\widetilde{MAE}\) for this combination is below 3.5%). \begin{table}
\begin{tabular}{l l c c c c c c c} \hline \hline \multicolumn{2}{c}{**Surrogate Models**} & \multicolumn{3}{c}{**Training**} & \multicolumn{3}{c}{**Verification**} \\  & \(R^{2}(z_{m})\) & \(R^{2}(z_{t})\) & **MAE** & **MAE** & **95\%** & \(\widetilde{MAE}\) & \(\widetilde{MAE}_{\sigma}\) & **Evaluation** \\ \hline \multirow{4}{*}{PRS} & \(\dot{m}_{cond\_3D}\) & & 0.94 & & 0.1596 & 0.139 & 0.0601 & 0.1851 & Good \\  & \(w_{cond\_3D}\) & & 0.95 & 0.98 & 0.1376 & 0.096 & 0.0415 & 0.1425 & Good \\  & \(\varphi\) & & 0.56 & 0.39 & 0.4587 & 0.2428 & 0.1044 & 0.4605 & Bad \\  & \(\Delta\) & & 0.95 & 0.98 & 0.0969 & 0.0793 & 0.0341 & 0.1001 & Good \\  & \(\eta\) & & 0.97 & 1.0 & 0.1274 & 0.0829 & 0.0356 & 0.1726 & Good \\ \hline \multirow{4}{*}{RBF} & \(\dot{m}_{cond\_3D}\) (Gauss.) * & & 1.0 & & 0.1824 & 0.1553 & 0.0688 & 0.1988 & Good \\  & \(w_{cond\_3D}\) (Gauss.) * & 1.0 & & 1.0 & 0.1078 & 0.0826 & 0.0355 & 0.1252 & Good \\  & \(\varphi\) (Multi-quad.) * & 1.0 & & 0.74 & 0.1413 & 0.1061 & 0.0457 & 0.1867 & Good \\  & \(\Delta\) (Cubic) * & 1.0 & & 1.0 & 0.2348 & 0.2052 & 0.0883 & 0.2566 & Moderate \\  & \(\eta\) (Multi-quad.) * & 1.0 & & 1.0 & 0.1601 & 0.0925 & 0.0398 & 0.2313 & Moderate \\ \hline \multirow{4}{*}{Kriging} & \(\dot{m}_{cond\_3D}\) & & 1.0 & & 0.2391 & 0.2176 & 0.0936 & 0.1964 & Good \\  & \(w_{cond\_3D}\) & & 1.0 & & 1.0 & 0.1409 & 0.1128 & 0.0473 & 0.1603 & Good \\  & \(\varphi\) & & 1.0 & & 0.85 & 0.1467 & 0.1066 & 0.0459 & 0.1916 & Good \\  & \(\Delta\) & & 1.0 & & 0.1616 & 0.1409 & 0.0606 & 0.1486 & Good \\  & \(\eta\) & & 1.0 & & 0.1171 & 0.0681 & 0.0293 & 0.1767 & Good \\ \hline \hline \end{tabular}  
* On each case is selected the best transformation type to obtain the greatest \(R^{2}\) evaluating the study variable.  
\end{table} 
Table 7: Evaluation of the surrogate models to predict water condensation for different metamodeled variables. Indeed, when applying this MF surrogate model, it is worth reminding that greater impeller damage is produced with greater condensation mass flow rates [70]. Finally, a graphical representation of the relation between the MF variables and the six significant input parameters regarded by ANOVA (see Section 5.2.2) is provided. Figures 8-10 show such 3D plots by considering pairs of parameters and leaving the omitted ones at their medium level (see Table 3). These figures represent low-fidelity, intermediate-variable, and high-fidelity features. Due to the negligible computational effort of the LF model employed in this work (see Section 4.2), the surface representation of 0D representations is obtained using an ad-hoc 100 \(\times\) 100 simulation campaign followed by linear interpolation. Intermediate figures showing differences are the graphical depictions of the corresponding polynomial response surface. Furthermore, Equation (17) allows us to obtain the metamodel of \(\dot{m}_{cond,3D}\) using the proposed MF strategy, represented in the predicted condensation.