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Artificial Neural Network Approximation of Fractional-Order Derivative Operators: Analysis and DSP Implementation

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Fractional Calculus and Fractional Differential Equations

Part of the book series: Trends in Mathematics ((TM))

Abstract

Fractional derivative operators, due to their infinite memory feature, are difficult to simulate and implement on software and hardware platforms. The available limited-memory approximation methods have certain lacunae, viz., numerical instability, ill-conditioned coefficients, etc. This chapter deals with the artificial neural network (ANN) approximation of fractional derivative operators. The input–output data of Gr\(\ddot{u}\)nwald–Letnikov and Caputo fractional derivatives for a variety of functions like ramp, power law type, sinusoidal, Mittag-Leffler functions is used for training multilayer ANNs. A range of fractional derivative order is considered. The Levenberg–Marquardt algorithm which is the extension of back-propagation algorithm is used for training the ANNs. The criterion of mean squared error between the outputs of actual derivative and the approximations is considered for validation. The trained ANNs are found to provide a very close approximation to the fractional derivatives. These approximations are also tested for the values of fractional derivative order which are not part of the training data-set. The approximations are also found to be computationally fast as compared to the numerical evaluation of fractional derivatives. A systematic analysis of the speedup achieved using the approximations is also carried out. Also the effect of increase in number of layers (net size) and the type of mathematical function considered on the mean squared error is studied. Furthermore, to prove the numerical stability and hardware suitability, the developed ANN approximations are implemented in real time on a DSP platform.

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Authors and Affiliations

  1. Department of Electronics Engineering, Ramrao Adik Institute of Technology, Navi Mumbai, India

    Pratik Kadam, Gaurav Datkhile & Vishwesh A. Vyawahare

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  1. Pratik Kadam
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  2. Gaurav Datkhile
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  3. Vishwesh A. Vyawahare
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Correspondence to Vishwesh A. Vyawahare .

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Editors and Affiliations

  1. Department of Mathematics, Savitribai Phule Pune University, Pune, Maharashtra, India

    Varsha Daftardar-Gejji

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Kadam, P., Datkhile, G., Vyawahare, V.A. (2019). Artificial Neural Network Approximation of Fractional-Order Derivative Operators: Analysis and DSP Implementation. In: Daftardar-Gejji, V. (eds) Fractional Calculus and Fractional Differential Equations. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-9227-6_6

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