Inferring the fractional nature of Wu Baleanu trajectories

Abstract

We infer the parameters of fractional discrete Wu-Baleanu time series by using machine learning architectures based on recurrent neural networks. Our results shed light on how clearly one can determine that a given trajectory comes from a specific fractional discrete dynamical system by estimating the fractional exponent and the scaling factor.

Anomalous Diffusion Characterization using Machine Learning Methods

“…MAEs for short trajectories are in fact considerably larger than for the rest of lengths for instance, more than 20000 thousand for length equal to 10 but around 6000 for lengths between 40 and 50. As it was also noticed in [9] the MAE slightly increases for trajectories with lengths between 40 and 50. It is worth to mention, that as the length increases, the number of trajectories in each bin decreases due to the bounds set for stopping the generation of trajectories (no term can be smaller than -1 nor greater than 3).…”

“…In other words, we want to see if they can determine if a delayed is incorporated to the model or not. To do so, we have used the data set described in this work jointly with the data set used in [9]. This second data set was also generated following the same procedure described in Algorithm 1 but with the following specifications.…”

“…We propose the architecture shown in Figure 2.4 to learn from time series, whatever their characteristics are. The definition of the architecture has been extracted from our paper [9], which we include in this Thesis.…”

“…Recently, we have studied how machine learning models can be used in connection with fractional models in order to infer the µ parameter and the scaling factor ν of the fractional version of the logistic equation [9]. In this work, we study up to which point we can predict the µ and ν parameters from short trajectories generated by the discrete fractional delayed dynamical system given by (8.2).…”

“…where µ := η − 1, see [9,134]. Our key observation -and starting point -in this article is that the term x n+1 − x n represents a discretization of the first order derivative, say u , inherited from the continuous model u (t) = µf (u(t)), t ≥ 0, (9.3) where f is a given real-valued function.…”