Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation

• A new stochastic intelligence method is developed to solve first Painlevé equation.
• Design of three unsupervised ANN models that satisfying exactly initial conditions.
• Optimization capability of SQP is exploited for training of design parameter of ANNs.
• Accuracy and convergence are validated in term of various performance criterions.
• Impact on effectiveness of the models is investigated by varying neurons in ANNs.

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.

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