## 1 Introduction  
The classical use of regression in physics, sometimes also referred to as non-linear fitting, is to try to determine \(d\) quantities \(y\in\mathbb{R}^{d}\) from a set of \(n\) measurements \(x\in\mathbb{R}^{q}\) with \(q\in\mathbb{N}\), using a theoretical mathematical model \(y=f(x,w)\) that depends on a certain number \(p\) of parameters \(w\in\mathbb{R}^{p}\). Typically, this is achieved by choosing the parameters \(w\) to minimize a selected error function, like the mean square error (MSE), with specific algorithms. To find the best solution for \(f\) is a classical optimization problem [1; 2; 3]. This method, however, fails to deliver stable and accurate results, for example, when the quantities \(y_{i}\) with \(i=1,...,d\) have different physical meanings and, consequently, depend on different components of the parameter vector \(w\) in fundamentally distinct ways. As a result, the mathematical model may be an insufficient approximation, may be too complex for a stable implementation or may be simply unknown [3]. An example where the usual multi-dimensional regression approach fails is in the determination of a substance from changes in its luminescence when several environmental conditions vary in an unknown and uncontrolled way. Luminescence quenching for oxygen detection represents a widespread application relevant in many fields like biomedical imaging, environmental monitoring, or process control [4] (see Section 4 for details). In this application, the quantity of interest is the concentration of molecular oxygen \([O_{2}]\). The measured quantity, either the luminescence intensity or luminescence intensity decay time of a special molecule (luminophore), is however, equally strongly dependent on the concentration \([O_{2}]\) and the temperature \(T\). As a result, it is difficult to extract two different physical quantities, namely, \([O_{2}]\) and \(T\), from the same set of data. Usually, \(T\) is measured separately with another device and given as an input to a mathematical model describing the dependency of those two quantities from the input data. The complexity increases further if more than one luminophore is present, and several parameters (e.g., \([O_{2}]\), \([CO_{2}]\), \(pH\)) have to be determined [5; 6; 7; 8; 9]. A possible method, which recently attracted great interest, is the use of feed-forward neural network (FFNN) architectures, with a certain number of hidden layers and an appropriate number of output neurons, each responsible for predicting the desired variables \(y_{i}\) with \(i=1,...,d\). In the example of oxygen sensing, the output layer would have a neuron for the oxygen concentration \([O_{2}]\) and one for the temperature \(T\). This work shows that, since the output neurons must use the same features (the output of the last hidden layer) for all variables [10; 11], FFNNs are insufficiently flexible. For the cases when the variables depend on fundamentally different ways from the inputs, this approach will give a result that is at best acceptable, and at worst unusable. This work proposes a new approach, which is based on multi-task learning (MTL) neural network architectures. This type of architectures are characterized by multiple branches of layers, that get their input from a common set of layers. This type of network can improve the model prediction performance by jointly learning correlated tasks [10; 11; 12; 13; 14]. In particular, the proposed MTL architectures are applied to the problem of luminescence quenching for oxygen sensing. Their performance in the prediction of oxygen concentration and temperature is analyzed and compared to that of a classical feed-forward neural network. In general, the proposed MTL approach may be of particular relevance in all those cases where the mathematical model \(y=f(x,w)\) is not known, too complex or not really of interest and the only goal of the regression problem is to build a system that is able to determine \(y\) as accurately as possible.