Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations

Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations

In this paper, novel computing paradigm by exploiting the strength of feed-forward artificial neural networks (ANNs) with Levenberg-Marquardt Method (LMM), and Bayesian Regularization Method (BRM) based backpropagation is presented to find the solutions of initial value problems (IVBs) of linear/non...

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Main Authors: Imtiaz Khan, Muhammad Asif Zahoor Raja, Muhammad Shoaib, Poom Kumam, Hussam Alrabaiah, Zahir Shah, Saeed Islam
Format: Article
Language:English
Published: IEEE 2020-01-01
Series:IEEE Access
Subjects:
Artificial neural networks
Levenberg-Marquardt method
Bayesian regularization method
nonlinear pantograph equation
regression analysis
intelligent computing
Online Access:https://ieeexplore.ieee.org/document/9154452/
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spelling doaj-29ec4f1924cd43c39841d58e1f5703872021-03-30T01:47:39ZengIEEEIEEE Access2169-35362020-01-01813791813793310.1109/ACCESS.2020.30118209154452Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential EquationsImtiaz Khan0Muhammad Asif Zahoor Raja1Muhammad Shoaib2Poom Kumam3https://orcid.org/0000-0002-5463-4581Hussam Alrabaiah4https://orcid.org/0000-0003-4597-5787Zahir Shah5https://orcid.org/0000-0002-5539-4225Saeed Islam6https://orcid.org/0000-0001-5263-4871Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, PakistanFuture Technology Research Center, National Yunlin University of Science and Technology, Douliu, TaiwanDepartment of Mathematics, COMSATS University Islamabad, Attock Campus, Attock, PakistanDepartment of Mathematics, KMUTT Fixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut&#x2019;s University of Technology Thonburi (KMUTT), Bangkok, ThailandCollege of Engineering, Al Ain University, Al Ain, United Arab EmiratesSCL 802 Fixed Point Laboratory, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), King Mongkut's University of Technology, Thonburi (KMUTT), Bangkok, ThailandDepartment of Mathematics, Abdul Wali Khan University Mardan, Mardan, PakistanIn this paper, novel computing paradigm by exploiting the strength of feed-forward artificial neural networks (ANNs) with Levenberg-Marquardt Method (LMM), and Bayesian Regularization Method (BRM) based backpropagation is presented to find the solutions of initial value problems (IVBs) of linear/nonlinear pantograph delay differential equations (LP/NP-DDEs). The dataset for training, testing and validation is created with reference to known standard solutions of LP/NP-DDEs. ANNs are implemented using the said dataset for approximate modeling of the system on mean squared error based merit functions, while learning of the adjustable parameters is conducted with efficacy of LMM (ANN-LMM) and BRMs (ANN-BRM). The performance of the designed algorithms ANN-LMM and ANN-BRM on IVPs of first, second and third order NP-FDEs are verified by attaining a good agreement with the available solutions having accuracy in the range from 10<sup>-5</sup> to 10<sup>-8</sup> and are further endorsed through error histograms and regression measures.https://ieeexplore.ieee.org/document/9154452/Artificial neural networksLevenberg-Marquardt methodBayesian regularization methodnonlinear pantograph equationregression analysisintelligent computing
collection DOAJ
language English
format Article
sources DOAJ
author Imtiaz Khan
Muhammad Asif Zahoor Raja
Muhammad Shoaib
Poom Kumam
Hussam Alrabaiah
Zahir Shah
Saeed Islam
spellingShingle Imtiaz Khan
Muhammad Asif Zahoor Raja
Muhammad Shoaib
Poom Kumam
Hussam Alrabaiah
Zahir Shah
Saeed Islam
Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations
IEEE Access
Artificial neural networks
Levenberg-Marquardt method
Bayesian regularization method
nonlinear pantograph equation
regression analysis
intelligent computing
author_facet Imtiaz Khan
Muhammad Asif Zahoor Raja
Muhammad Shoaib
Poom Kumam
Hussam Alrabaiah
Zahir Shah
Saeed Islam
author_sort Imtiaz Khan
title Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations
title_short Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations
title_full Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations
title_fullStr Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations
title_full_unstemmed Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations
title_sort design of neural network with levenberg-marquardt and bayesian regularization backpropagation for solving pantograph delay differential equations
publisher IEEE
series IEEE Access
issn 2169-3536
publishDate 2020-01-01
description In this paper, novel computing paradigm by exploiting the strength of feed-forward artificial neural networks (ANNs) with Levenberg-Marquardt Method (LMM), and Bayesian Regularization Method (BRM) based backpropagation is presented to find the solutions of initial value problems (IVBs) of linear/nonlinear pantograph delay differential equations (LP/NP-DDEs). The dataset for training, testing and validation is created with reference to known standard solutions of LP/NP-DDEs. ANNs are implemented using the said dataset for approximate modeling of the system on mean squared error based merit functions, while learning of the adjustable parameters is conducted with efficacy of LMM (ANN-LMM) and BRMs (ANN-BRM). The performance of the designed algorithms ANN-LMM and ANN-BRM on IVPs of first, second and third order NP-FDEs are verified by attaining a good agreement with the available solutions having accuracy in the range from 10<sup>-5</sup> to 10<sup>-8</sup> and are further endorsed through error histograms and regression measures.
topic Artificial neural networks
Levenberg-Marquardt method
Bayesian regularization method
nonlinear pantograph equation
regression analysis
intelligent computing
url https://ieeexplore.ieee.org/document/9154452/
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AT muhammadasifzahoorraja designofneuralnetworkwithlevenbergmarquardtandbayesianregularizationbackpropagationforsolvingpantographdelaydifferentialequations
AT muhammadshoaib designofneuralnetworkwithlevenbergmarquardtandbayesianregularizationbackpropagationforsolvingpantographdelaydifferentialequations
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