## 5 Results and Discussion
The results for \(AE_{[O_{2}]}\) and \(AE_{T}\) for the three variations of FFNN A are shown in Figures 7 and 8, respectively. From Figures 7 and 8 can clearly be seen that increasing the number of neurons helps at the beginning. A further increase in \(\hat{n}\) does not produce an improvement in prediction quality, on the contrary it gets worse. These results indicate that this simple FFNN can extract at the same time the two quantities with an accuracy which is at best poor and at worst unusable. Networks B and C try to address this problem by adding, as described in previous sections, respectively one and two branches after the last hidden layer in network A. The results of the prediction from the networks B and C are then compared to those from network A with \(\hat{n}=50\). Figure 9 shows the calculated \(AE_{[O_{2}]}\) for the three networks for the same \([O_{2}]\) intervals as before as a box plot, where the median is visible as a red line. As it can be seen from Figure 9, the error in the prediction of network B is similar to that of network A. However, \(AE_{[O_{2}]}\) is significantly improved when using network C. The additional branch in network C compared to network B clearly make the predictions much more accurate and, more importantly, much less spread around the median. Figure 8: Kernel density estimation for \(AE_{T}\) for network A. Left: 30 neurons per hidden layer; middle: 50 neurons per hidden layer, right: 80 neurons per hidden layer. Figure 7: Kernel density estimation for \(AE_{[O_{2}]}\) with network A. Left: 30 neurons per hidden layer; middle: 50 neurons per hidden layer, right: 80 neurons per hidden layer. The distribution of the \(AE_{[O_{2}]}\) is better illustrated by plotting the KDE (Figure 10). The results indicate that the distribution assumes much smaller values and is peaked around zero for network C, in contrast with network A and B that have a quite wide tail that propagates toward higher values, reaching values as high as 10% air for network A and 8% air for network B. Finally, the results of the same analysis for the prediction of the temperature are shown in Figure 11. Here the calculated \(AE_{T}\) for the same three networks is shown as a box plot, where the median is visible as a red line. Figure 11: Absolute error in the prediction of the temperature using network A, B, and C. Left: Network A with 50 neurons per hidden layer; middle: network B, right: network C.  
Figure 9: Absolute error in the prediction of the \(O_{2}\) concentration for the different concentration ranges using network A, B, and C. Left: Network A with 50 neurons per hidden layer; middle: network B, right: network C.  
As it can be seen from Figure 11, \(AE_{T}\) is much more concentrated around the median when using network C. These results indicate that the prediction of the temperature is substantially improved when using this network. The distribution of the \(AE_{T}\) using the KDE is shown in Figure 12. Thanks to the additional task-specific hidden layer of the network C compared to network B, the KDE is higher and peaked around zero, with practically no contributions above 5 \({}^{\circ}\)C.  
Finally, the performance of the three neural networks are be summarized by calculating the \(MAE\) as defined in Equation (6) for the oxygen concentration and the temperature prediction. The results are listed in Table 2. The network C outperforms all the other networks analyzed in predicting both \([O_{2}]\) and \(T\), achieving a mean absolute error of only 0.5% air for the oxygen concentration and of 2.2 \({}^{\circ}\)C for the temperature. The results of Table 2 show that a simple FFNN as network A is not suitable to extract the two quantities of interest at the same time with good accuracy, since it is not flexible enough. The reason is that the two predicted quantities will depend on the same set of features generated by the hidden layers of network A.