Abstract
The classical approach to non-linear regression in physics is to take a mathematical model describing the functional dependence of the dependent variable from a set of independent variables, and then using non-linear fitting algorithms, extract the parameters used in the modeling. Particularly challenging are real systems, characterized by several additional influencing factors related to specific components, like electronics or optical parts. In such cases, to make the model reproduce the data, empirically determined terms are built in the models to compensate for the difficulty of modeling things that are, by construction, difficult to model. A new approach to solve this issue is to use neural networks, particularly feed-forward architectures with a sufficient number of hidden layers and an appropriate number of output neurons, each responsible for predicting the desired variables. Unfortunately, feed-forward neural networks (FFNNs) usually perform less efficiently when applied to multi-dimensional regression problems, that is when they are required to predict simultaneously multiple variables that depend from the input dataset in fundamentally different ways. To address this problem, we propose multi-task learning (MTL) architectures. These are characterized by multiple branches of task-specific layers, which have as input the output of a common set of layers. To demonstrate the power of this approach for multi-dimensional regression, the method is applied to luminescence sensing. Here, the MTL architecture allows predicting multiple parameters, the oxygen concentration and temperature, from a single set of measurements.
Highlights
The classical use of regression in physics, sometimes referred to as non-linear fitting, is to try to determine d quantities y ∈ Rd from a set of n measurements x ∈ Rq with q ∈ N, using a theoretical mathematical model y = f ( x, w) that depends on a certain number p of parameters w ∈ R p
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
Summary
The classical use of regression in physics, sometimes referred to as non-linear fitting, is to try to determine d quantities y ∈ Rd from a set of n measurements x ∈ Rq with q ∈ N, using a theoretical mathematical model y = f ( x, w) that depends on a certain number p of parameters w ∈ R p. This is achieved by choosing the parameters w to minimize a selected error function, like the mean square error (MSE), with specific algorithms.
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