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

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Abstract

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.

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Author information

Authors and Affiliations

  1. Department of Electronic Engineering, International Islamic University, Islamabad, Pakistan

    Muhammad Asif Zahoor Raja

  2. Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock, Pakistan

    Muhammad Asif Zahoor Raja

  3. Pakistan Institute of Engineering and Applied Science, Nilore, Islamabad, Pakistan

    Siraj-ul-Islam Ahmad

  4. Mohammad Ali Jinnah University, Islamabad, Pakistan

    Raza Samar

Authors
  1. Muhammad Asif Zahoor Raja
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  2. Siraj-ul-Islam Ahmad
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  3. Raza Samar
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Corresponding author

Correspondence to Muhammad Asif Zahoor Raja.

Appendix

Appendix

A set of optimal weights trained by different optimization algorithms is provided in Tables 9, 10 and 11 for the cases λ = 0.5, 1.0 and 2, respectively, of the transformed Bratu boundary value problem.

Table 9 Parameters obtained for the DENN network for λ = 0.5
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Table 10 Parameters obtained for the DENN network for λ = 1
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Table 11 Parameters obtained for the DENN network for λ = 2
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Raja, M.A.Z., Ahmad, SuI. & Samar, R. Neural network optimized with evolutionary computing technique for solving the 2-dimensional Bratu problem. Neural Comput & Applic 23, 2199–2210 (2013). https://doi.org/10.1007/s00521-012-1170-4

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