Single layer Chebyshev neural network model with regression-based weights for solving nonlinear ordinary differential eq
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RESEARCH PAPER
Single layer Chebyshev neural network model with regression‑based weights for solving nonlinear ordinary differential equations S. Chakraverty1 · Susmita Mall1 Received: 25 July 2019 / Revised: 15 January 2020 / Accepted: 5 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this investigation, a novel single layer Functional Link Neural Network namely, Chebyshev artificial neural network (ChANN) model with regression-based weights has been developed to handle ordinary differential equations. In ChANN, the hidden layer is removed by an artificial expansion block of the input patterns by using Chebyshev polynomials. Thus the technique is more effectual than the multilayer ANN. Initial weights from the input layer to the output layer are taken by a regression-based model. Here, feed-forward structure and back-propagation algorithm of the unsupervised version have been utilized to make the error values minimal. Numerical examples and comparisons with other methods exhibit the superior behavior of this technique. Keywords Chebyshev neural network · Regression based weights · Error back propagation algorithm · Ordinary differential equations
1 Introduction Differential equations appear in modeling innumerable phenomena that apply in different fields of science and technology. Various real-world problems of mathematics, defense, physics, engineering, etc. may be described by ODEs and PDEs [1–5]. In most of the cases, we cannot find an exact solution of differential equations easily. Thus various types of numerical techniques viz. Runge–Kutta, finite difference, predictor–corrector and finite volume etc. [6–10] have been employed to solve these equations. In recent years, machine learning algorithms viz. artificial neural networks (ANNs) approach has attracted much consideration to its advantages such as learning, adaptive, error computation, fault-tolerance etc. Currently, ANN models are being used to solve ODEs and PDEs. The ANN based solutions for differential equations are more advantageous compared to other traditional methods. The computational complexity does not increase considerably with the number * S. Chakraverty [email protected] Susmita Mall [email protected] 1
Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha 769008, India
of sampling points. Moreover, other numerical methods are usually iterative in nature. ANN may be one of the reliefs where we may overcome this repetition of iterations. After training of the ANN model, we may use it as a black box to get numerical results at any arbitrary point in the domain. ANN based approaches are also utilized by many authors for solving DEs. As such below we mention a few of them. Lee and Kang [11] developed Hopfield neural network model for solving first order ODEs. Meade and Fernandez [12, 13] proposed ANN and B1-splines methods for solving linear and non-linear ODEs. Lagaris et al. [14] approached multi-layer ANN architecture and BFGS optimization method to
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