## 6 Conclusions  
In this work, different neural networks architectures were investigated to solve the problem of extracting multiple separate physical quantities at the same time from a single dataset. This type of multi-dimensional regression problems in physics can be challenging or impossible to solve if the mathematical models describing the functional dependence of the dependent variable from a set of independent variables are too complex or unknown. The proposed approach consists in using neural network MTL architectures, which are characterized by a common set of layers and then task-specific layers for each quantity to be determined. Thanks to the additional task-specific hidden layers, this type of networks can be trained to perform better than conventional FFNNs when the quantities to be predicted are characterized by a significant difference in physical behavior.  
The approach is demonstrated by applying it to an oxygen luminescence sensing application. The conventional methods rely on a separate temperature determination which is then used as input to correct the extraction of the oxygen concentration from a dataset. This work demonstrates how it is possible to extract from a single dataset of phase shift measurements both the oxygen concentration and the temperature of the medium. The distributions of \(AE_{[O_{2}]}\) and \(AE_{T}\) are significantly narrower and much more concentrated around zero with the proposed MTL network (type C), as compared to FFNNs without specific and dedicated layers for each \([O_{2}]\) and \(T\). With the latter network the predictions are only based on common features (the ones generated by the common layers) that fail to be flexible enough to describe both \([O_{2}]\) and \(T\). The results indicate that from one single measurement, it is possible to determine two physically different quantities, one of which is dependent from the other. To the best of the authors' knowledge, this is the first time that more than one parameter (here \([O_{2}]\) and \(T\)) are extracted using a single luminophore by a single measurement channel under constant conditions. The implication is that a sensor using the proposed approach could be able to extract much more information from the measurements than one based on conventional analytical modeling.  
This work aimed to open the road to new ways of extracting multiple physical quantities from a common set of data at the same time to achieve consistent results that are both accurate and  
\begin{table}
\begin{tabular}{c c c c c} \hline \(\alpha_{1}\) & \(\alpha_{2}\) & \(\alpha_{3}\) & \(MAE_{[O_{2}]}\) & \(MAE_{[T]}\) \\ \hline
0.3 & 5.0 & 5.0 & 0.54\% air & 2.2 \({}^{\circ}\)C \\
0.3 & 5.0 & 15.0 & 0.61\% air & 2.35 \({}^{\circ}\)C \\
0.3 & 5.0 & 25.0 & 0.89\% air & 2.32 \({}^{\circ}\)C \\
0.3 & 1.0 & 5.0 & 0.58\% air & 2.25 \({}^{\circ}\)C \\
0.3 & 15.0 & 5.0 & 0.94\% air & 2.67 \({}^{\circ}\)C \\
0.3 & 25.0 & 5.0 & 0.96\% air & 2.55 \({}^{\circ}\)C \\ \hline \end{tabular}
\end{table}
Table 3: Summary of the performance for network C with various loss weights.  
stable. The described approach is relevant for many practical applications in sensor science and demonstrates that MTL architectures have the potential of revolutionizing the approach to non-linear multi-dimensional regression.  
Conceptualization, U.M. and F.V.; methodology, U.M. and F.V.; software, U.M.; writing, U.M. and F.V.; physics model and examples, F.V.  
This research received no external funding.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
FFNN Feed-forward neural networks  
MTL Multi-task learning  
MSE Mean square error  
AE Absolute error  
MAE Mean average error  
KDE Kernel density estimate