Neural Approximators for Variable-Order Fractional Calculus Operators (VO-FC)

Bartosz Puchalski

doi:10.1109/access.2022.3143893

Neural Approximators for Variable-Order Fractional Calculus Operators (VO-FC)

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-Letnikow definition of variable-order fractional operators suitable for 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. 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.