Siraj-ul-Islam Ahmad

Numerical treatment for solving one-dimensional Bratu problem using neural networks

In this paper, numerical treatment is presented for the solution of boundary value problems of one-dimensional Bratu-type equations using artificial neural networks. Three types of transfer functions including Log-sigmoid, radial basis, and tan-sigmoid are used in the neural networks’ modeling. The optimum weights for all the three networks are searched with the interior point method. Various test cases of Bratu-type equations have been simulated using the developed models. The accuracy, convergence, and effectiveness of the methods are substantiated by a large number of simulation data for each model by taking enough independent runs.

Neural network optimized with evolutionary computing technique for solving the 2-dimensional Bratu problem

In this paper, a stochastic technique is developed to solve 2-dimensional Bratu equations using feed-forward artificial neural networks, optimized with genetic and interior-point algorithms. The 2-dimensional equations are first transformed into a 1-dimensional boundary value problem, and a mathematical model of the transformed equation is then formulated with neural networks using an unsupervised error. Network weights are optimized to minimize the error. Evolutionary computing based on genetic algorithms is used as a tool for global search, integrated with an interior-point method for rapid local convergence. The methodology is applied to solve three cases of boundary value problems for the Bratu equations. The accuracy, convergence and effectiveness of the scheme is validated for a large number of simulations. Comparison of results is made with the exact solution derived using MATHEMATICA, and is found to be in good agreement.

A new stochastic technique for painlevé equation-I using neural network optimized with swarm intelligence

A methodology for solution of Painlevé equation-I is presented using computational intelligence technique based on neural networks and particle swarm optimization hybridized with active set algorithm. The mathematical model of the equation is developed with the help of linear combination of feed-forward artificial neural networks that define the unsupervised error of the model. This error is minimized subject to the availability of appropriate weights of the networks. The learning of the weights is carried out using particle swarm optimization algorithm used as a tool for viable global search method, hybridized with active set algorithm for rapid local convergence. The accuracy, convergence rate, and computational complexity of the scheme are analyzed based on large number of independents runs and their comprehensive statistical analysis. The comparative studies of the results obtained are made with MATHEMATICA solutions, as well as, with variational iteration method and homotopy perturbation method.

Numerical Treatment for Painlevé Equation I Using Neural Networks and Stochastic Solvers

In this chapter, a new stochastic numerical treatment is presented for solving Painleve I equation. The mathematical model of the equation is formulated with feed-forward artificial neural networks. Linear combination of the networks defines the unsupervised error for the equation. The error is reduced subject to the availability of appropriate weights of networks. Training of weights is done with genetic algorithm, simulating annealing and pattern search algorithms hybridized with interior point algorithm for rapid local search. The reliability and effectiveness is validated with the help of statistical analysis. Comparison of results is made with standard approximate analytic solvers of the equation. It is found that the proposed results are in a good agreement with their corresponding numerical solutions.