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Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation
Affiliation:1. Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock, Pakistan;2. Faculty of Engineering Science and Technology, Hamdard University, Islamabad Campus, Pakistan;3. Department of Mathematics, Pennsylvania State University, York Campus, York, PA 17403-3398, USA;4. Department of Computer Science, USTHB, BP32 El Alia, Bab Ezzouar, 16111 Algiers, Algeria;5. Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan;6. Department of Electrical Engineering, Mohammad Ali Jinnah University, Islamabad, Pakistan;1. Department of Computer Science, Palacky University, 17. listopadu 12, Olomouc, Czech Republic;2. Department of Electrical Engineering, RIMT Institute, Mandi Gobindgarh 147301, India;3. School of Mathematics and Computer Applications, Thapar University, Patiala 147004, India;1. Computer Science, Faculty of Computers and Informatics, Suez Canal University, Egypt;2. National Authority of Remote Sensing and Space Sciences, Cairo, Egypt;1. Collaborative Innovation Center of Electric Vehicles in Beijing, Beijing Institute of Technology, Beijing, 100081, China;2. JIANGLU Machinery & Electronics Group Co, LTD, Xiangtan, 430300, China
Abstract:In this paper, novel computing approach using three different models of feed-forward artificial neural networks (ANNs) are presented for the solution of initial value problem (IVP) based on first Painlevé equation. These mathematical models of ANNs are developed in an unsupervised manner with capability to satisfy the initial conditions exactly using log-sigmoid, radial basis and tan-sigmoid transfer functions in hidden layers to approximate the solution of the problem. The training of design parameters in each model is performed with sequential quadratic programming technique. The accuracy, convergence and effectiveness of the proposed schemes are evaluated on the basis of the results of statistical analyses through sufficient large number of independent runs with different number of neurons in each model as well. The comparisons of these results of proposed schemes with standard numerical and analytical solutions validate the correctness of the design models.
Keywords:Painlevé transcendents  Neural network  Sequential quadratic programming  Nonlinear differential equations  Activation functions
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