## 5 Results and Discussion
When network A tries to learn better weights to predict, for example, the temperature, these will, however, influence also the \([O_{2}]\) prediction and vice-versa. So the common set of weights that are learned can not be optimized for each quantity separately at the same time. The MTL network B tries to address this problem with a separate branch of task-specific layers for \([O_{2}]\). The tests show however that this architecture is only marginally better for the prediction of \([O_{2}]\) and even worse for the prediction of \(T\). This is probably due an insufficient flexibility of the network and shows that even if only one parameter were of interest, e.g., \([O_{2}]\) one single additional branch is not sufficient. A significant improvement is achieved with the MTL network C: the two task-specific branches give the network the flexibility of learning a set of weights (the ones in the branches) specific to each quantity, therefore achieving exceptionally good predictions on both \([O_{2}]\) and \(T\). Note that in this work the hyper-parameter tuning [15] for each network was not performed since the goal is not to achieve the lowest possible \(MAE\)s but rather to demonstrate the advantages and potential of MTL compared to classical FFNN approaches. For the implementation in a measuring instrument, therefore, a further phase of parameter tuning specifically dependent on the application would be needed. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Network** & \(MAE_{[O_{2}]}\) & \(MAE_{[T]}\) \\ \hline Network A (\(\hat{n}=30\)) & 6.0\% air & 9.3 \({}^{\circ}\)C \\ Network A (\(\hat{n}=50\)) & 1.7\% air & 3.3 \({}^{\circ}\)C \\ Network A (\(\hat{n}=80\)) & 2.3\% air & 4.3 \({}^{\circ}\)C \\ Network B & 1.5\% air & 6.5 \({}^{\circ}\)C \\ Network C & 0.5\% air & 2.2 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the performance for the three types of neural networks. Figure 12: Kernel density estimation for \(AE_{T}\) for networks A (left), B (middle), and C (right). An interesting question is what is the mutual influence of the branches in network C when the loss weights \(\alpha_{i}\) are varied. To answer this question, a study was performed with various values of the global cost function weights. The results are shown in Table 3.  
By increasing progressively the weight for the temperature branch, \(\alpha_{3}\), the MAE\({}_{[T]}\) is not reduced further and appears rather insensitive to \(\alpha_{3}\). However, \(MAE_{[O_{2}]}\) increases slightly, since the higher values of \(\alpha_{3}\) shift the relative importance of the tasks the network is trying to learn. Increasing the weight for the oxygen branch \(\alpha_{2}\) negatively affects the oxygen prediction since \(MAE_{[O_{2}]}\) increases slightly. The reason why this is happening is that the \(\alpha_{2}\) is becoming much bigger than \(\alpha_{1}\). This shows that for the prediction of the oxygen concentration both the branches predicting \(T\) and \([O_{2}]\) at the same time and the one predicting \([O_{2}]\) are important: neglecting one will make the other works less efficiently. The temperature, on the other had, is predicted almost with the same kind of accuracy independently of the weights \(\alpha_{2}\), indicating that the temperature branch is not dependent from the \([O_{2}]\) branch.