## 5 Results and Discussion  
As described in Section 4, the applied problem investigated in this work is a complex one since the two quantities to be extracted from the data (\([O_{2}]\) and \(T\)) depend from the input in different ways. It is, therefore, not obvious that is possible to build a model which is able to predict both \([O_{2}]\) and \(T\) at the same time with good accuracy. The fist network investigated is the simple FFNN A described in Section 3.1. For this network, the number of neurons was progressively increased (\(\hat{n}=10,30,50,80\)) to study how \(AE_{[O_{2}]}\) and \(AE_{T}\) are affected by an increasingly complex network and to determine if it is possible to obtain a good prediction. The calculated \(AE_{[O_{2}]}\) for different \(O_{2}\) concentrations were grouped in bins of 10% air for a clearer illustration and are shown in Figure 5 as a box plot, where the median is visible as a red line. In all the boxplots in this paper the central box is the interquartile range and contains the middle 50% of the results, while the whiskers indicates the minimum and maximum of all the data [32]. As can be seen in Figure 5, the results are quite poor if \(\hat{n}=30\) (results for \(\hat{n}=10\) are comparable to those with \(\hat{n}=30\) and are not shown here). \(AE_{[O_{2}]}\) can assume values as big as 18% air, with a broad distribution. Increasing the number of neurons in the hidden layers to \(\hat{n}=50\) improves the prediction, reducing both the median and the distribution. A further increase to \(\hat{n}=80\), however, does not result in better a prediction, showing the limits of this architecture to capture the details of the physical system. The results for the prediction of the temperature for the same three networks are shown in Figure 6. Also \(AE_{T}\) improves initially by increasing the number of neurons to \(\hat{n}=50\), but does not get any better when the number of neurons is further increased to \(\hat{n}=80\). The boxplots of Figures 5 and 6 show that \(AE_{[O_{2}]}\) and \(AE_{T}\) can assume quite high values, therefore demonstrating how the model is not really able to make a prediction with an accuracy that may be used in any commercial application. The performance of the three FFNN of type A can be summarized calculating the \(MAE\) as defined in Equation (6). The results are listed in Table 1. Consistently with what previously observed for the absolute error, the best network performance is obtained with \(\hat{n}=50\), achieving a mean absolute error of \(MAE_{[O_{2}]}=1.7\%\) air and \(MAE_{T}=3.3\)\({}^{\circ}\)C. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\hat{n}\) & \(MAE_{[O_{2}]}\) & \(MAE_{[T]}\) \\ \hline
30 & 6.0\% air & 9.3\({}^{\circ}\)C \\
50 & 1.7\% air & 3.3\({}^{\circ}\)C \\
80 & 2.3\% air & 4.3\({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the performance for the FFNNs A.  
Figure 5: Absolute error \(AE_{[O_{2}]}\) in the prediction of the \(O_{2}\) concentration for the different concentration ranges using network A. Left: 30 neurons per hidden layer; middle: 50 neurons per hidden layer, right: 80 neurons per hidden layer. Figure 6: Absolute error \(AE_{T}\) in the prediction of \(T\) for the different temperatures using network A. Left: 30 neurons per hidden layer; middle: 50 neurons per hidden layer, right: 80 neurons per hidden layer. For a practical application, the probability density distributions of the \(AE\)s for both parameters represent a much fundamental quantity since it carries information on the probability of the network to predict the expected value. For this reason, the kernel density estimate (KDE) of the distributions of the \(AE\)s was used for analysis. KDE is a non-parametric algorithm to estimate the probability density function of a random variable by inferring the population distribution based on a finite data sample [33]. For the plots we have used a Gaussian Kernel and a Scott bandwidth estimation using the seaborn Python package [34].