## 3 Results
### Neural Network Development  
For all the network configurations discussed hereinabove, performance indexes are summarized in Table 4. L1 architecture yields poor results. ANN1, ANN2 and ANN3 show a high RMS over a 1000-sample population. On the contrary, for all the other networks configurations featuring either L2 (3 HLs) or L3 architecture (4 HLs), RMS is in the neighbourhood of 1%. The increase up to 4 HLs does not produce any further performance enhancement. Among the investigated ANN configurations with 3 HLs, the sensitivity analysis performed on neurons number ends in the identification of ANN5 as most performing network (0.67% RMS in the test). For this reason, ANN5 is chosen to build the final simulation tool for SOFC performance. On the other hand, ANN4 scores the worst performance (1.21% _RMS_ in the test) among the 3-HL configurations. ANN4 and ANN5 error dispersion during the last epoch training phase is shown in Figures 3b and 3c, respectively).  
Figure 2: Performances maps at constant current load: 100 mA\(\cdot\)cm\({}^{-2}\) (**a**) and 250 mA\(\cdot\)cm\({}^{-2}\) (**b**).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**ANN** & **Training** & \multicolumn{2}{c}{**Validation**} & \multicolumn{2}{c}{**Test**} & \\ \hline
**ID** & **RMS \%** & **MSE** & **RMS \%** & **RMS \%** & **Embeds** \\ \hline ANN1 & 10.75\% & 0.0117 & 10.8\% & 10.97\% & 42 \\ ANN2 & 12.92\% & 0.0169 & 13.0\% & 12.86\% & 31,000 \\ ANN3 & 13.02\% & 0.0171 & 13.1\% & 13.01\% & 100,000 \\ ANN4 & 1.36\% & 0.0002 & 1.4\% & 1.21\% & 10,600 \\ ANN5 & 0.80\% & \textless{}0.0001 & 0.7\% & 0.67\% & 120,000 \\ ANN6 & 0.99\% & \textless{}0.0001 & 1.0\% & 1.05\% & 100,000 \\ ANN7 & 1.01\% & 0.0001 & 1.0\% & 1.12\% & 100,000 \\ ANN8 & 0.98\% & \textless{}0.0001 & 1.0\% & 1.07\% & 100,000 \\ ANN9 & 1.10\% & 0.0001 & 1.1\% & 1.05\% & 100,000 \\ ANN 10 & 0.75\% & \textless{}0.0001 & 0.8\% & 0.78\% & 112,000 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Artificial neural networks performances.  
Figure 3: Training of ANN2 (**a**), ANN4 (**b**) and ANN5 (**c**): error dispersion.  
With regard the validation data-set of the algorithm, more interesting comments can be made. The error dispersion obtained throughout the validation phase is plotted in Figure 4a for ANN2, Figure 4b for ANN4 and Figure 4c for ANN5.With regard to the simulated and experimental voltage data, it is clear as ANN2 is not able to reproduce the SOFC performance varying all the operating conditions input of the developed model. Moreover, observing the error dispersion trend the better generalization capability of ANN5 with respect to ANN4 is evident.  
Finally, on the basis of an independent data-subset excluded from the training and validation phase, the simulation performances of the neural network under investigation are test. Once again, simple configuration with just 2 HLs produced output results affected with high error, ANN5 shows the lowest \(RMS\) (0.67%). Figure 5 depicts the error dispersion for ANN2, ANN4 and ANN5 highlighting the best performance of ANN5 in reproducing the SOFC behaviour.  
Figure 4: Validation of ANN2 (**a**), ANN4 (**b**) and ANN5 (**c**): error dispersion.