Abstract
The paper presents research on the approximation of variable-order fractional operators by recurrent neural networks. The research focuses on two basic variable-order fractional operators, i.e., integrator and differentiator. The study includes variations of the order of each fractional operator. The recurrent neural network architecture based on GRU (Gated Recurrent Unit) cells functioned as a neural approximation for selected fractional operators. The paper investigates the impact of the number of neurons in the hidden layer, treated as a hyperparameter, on the quality of modeling error. Training of the established recurrent neural network was performed on synthetic data sets. Data for training was prepared based on the modified Grünwald-Letnikov definition of variable-order fractional operators suitable for convenient numerical computing without memory effects. The research presented in this paper showed that recurrent network architecture based on GRU-type cells can satisfactorily approximate targeted simple yet functional variable-order fractional operators with minor modeling errors. In addition, the research also compares the presented solution with basic and recurrent neural networks that utilize Tapped Delay Lines (TDL) in their structure. The presented solution is a novel approach to the approximation of VO-FC operators. It has the advantage of automatic selection of neural approximator parameters by optimization based on data customized for specific requirements.
Highlights
Fractional order calculus [1] has been known since the 17th century
This paper extends the previous research towards modeling approximations of variable order fractional calculus operator (VO-FC) operators based on recurrent neural networks
Based on the results presented it can be concluded that the proposed GRU cell-based recurrent neural network architectures approximate VO-FC operators based on Grünwald-Letnikov definition without memory effects with excellent overall performance
Summary
Fractional order calculus [1] has been known since the 17th century This field of science concerns various methods that lead to a generalization of integration and differentiation operators to real or complex orders. In the last few decades, the interest in fractional-order calculus has grown in a very significant way in both the research and engineering fields, with special attention in modeling and simulation of physical phenomena as reported in [2] and [3]. Used in the above exemplary research and engineering areas provide a specific way to generalize mathematical models by introducing an arbitrary order, for instance, denoted as α into the integro-differential fractional operator Dα, where α ∈ R or in general case α ∈ C. Cases for which α ∈ R will be considered
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