1. Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations
- Author
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Muhammad Asif Zahoor Raja, Poom Kumam, Imtiaz Khan, Muhammad Shoaib, Hussam Alrabaiah, Zahir Shah, and Saeed Islam
- Subjects
0209 industrial biotechnology ,General Computer Science ,Mean squared error ,Differential equation ,Computer science ,Computer Science::Neural and Evolutionary Computation ,02 engineering and technology ,Bayesian interpretation of regularization ,regression analysis ,020901 industrial engineering & automation ,0202 electrical engineering, electronic engineering, information engineering ,Initial value problem ,General Materials Science ,Bayesian regularization method ,Artificial neural networks ,Artificial neural network ,nonlinear pantograph equation ,General Engineering ,Delay differential equation ,Backpropagation ,Levenberg–Marquardt algorithm ,Nonlinear system ,Levenberg-Marquardt method ,020201 artificial intelligence & image processing ,lcsh:Electrical engineering. Electronics. Nuclear engineering ,intelligent computing ,lcsh:TK1-9971 ,Algorithm - Abstract
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 10-5 to 10-8 and are further endorsed through error histograms and regression measures.
- Published
- 2020