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.
Highlights
A particular form of functional differential equations that involve proportional delays are called pantograph or generalized pantograph equations
A novel design of two-layer feed-forward artificial neural networks (ANNs) backpropagated with Levenberg-Marquardt method (LMM), i.e., ANN-Levenberg-Marquardt Method (LMM) and Bayesian Regularization Method (BRM), i.e., ANNBRM is presented for finding the solution of initial value problems (IVBs) of LP/NP DDEs
The results show that there is no noticeable difference in the performance for LP-DDE while NP-DDE the small error values are attained by ANN-RBM
Summary
A particular form of functional differential equations that involve proportional delays are called pantograph or generalized pantograph equations. Applied AI techniques through Levenberg-Marquardt Method (LMM) and Bayesian Regularization Method (BRM) based backpropagation of neural networks to solve initial value problems (IVBs) of linear/nonlinear pantograph delay differential equations (LP/NP-DDEs). This encourages the authors to investigate an AI technique to solve IVPs of LP/NPDDEs represented in equations (1-3). A novel design of two-layer feed-forward artificial neural networks (ANNs) backpropagated with Levenberg-Marquardt method (LMM), i.e., ANN-LMM and Bayesian Regularization Method (BRM), i.e., ANNBRM is presented for finding the solution of IVBs of LP/NP DDEs. The outcomes of the proposed scheme are summarized in the form of concluding remarks along with provision of future related studies
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