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Design of artificial neural network models optimized with sequential quadratic programming to study the dynamics of nonlinear Troesch’s problem arising in plasma physics

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Abstract

In this study, a computational intelligence technique based on three different designs of artificial neural networks (ANNs) is presented to solve the nonlinear Troesch’s boundary value problem arising in plasma physics. The structure of three ANN models is formulated by incorporating log-sigmoid (ANN-LS), radial-base (ANN-RB) and tan-sigmoid (ANN-TS) kernel functions in the hidden layers. Mathematical modeling of the problem is constructed for all three feed-forward ANN models by defining an error function in an unsupervised manner. Sequential quadratic programming method is employed for the learning of unknown parameters for all three ANN-LS, ANN-RB and ANN-TS schemes. The proposed models are applied to solve variants of Troesch’s problems by taking the small and large values of critical parameter in the system. A comparison of the proposed solution obtained from these models has been made with the standard numerical results of Adams method. The accuracy and convergence of the proposed models are investigated through results of statistical analysis in terms of performance indices based on the mean absolute deviation, root-mean-square error and variance account for.

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Author information

Authors and Affiliations

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

    Muhammad Asif Zahoor Raja & Fiaz Hussain Shah

  2. Department of Basic Sciences, RIPHAH International University, Islamabad, Pakistan

    Muhammad Tariq

  3. Department of Mathematics, University of Gujrat, Gujrat, Pakistan

    Iftikhar Ahmad

  4. Department of Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan

    Siraj ul Islam Ahmad

Authors
  1. Muhammad Asif Zahoor Raja
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  2. Fiaz Hussain Shah
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Correspondence to Muhammad Asif Zahoor Raja.

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Appendix

Appendix

The values of weights for SQP-ANN-LS, SQP-ANN-RB and SQP-ANN-TS are given in respective Tables 7, 8 and 9 for scenario 1 and in respective Tables 10, 11, and 12 for scenario 2 of Troesch’s problem.

Table 7 A set of best weights of ANN-LS for each case of scenario 1 of Troesch’s problem
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Table 8 A set of best weights of ANN-RB for each case of scenario 1 of Troesch’s problem
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Table 9 A set of trained weights of ANN-TS for each case of scenario 1 of Troesch’s problem
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Table 10 A set of trained weights of ANN-LS for each case of scenario 2 of Troesch’s problem
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Table 11 A set of trained weights of ANN-RB for each case of scenario 2 of Troesch’s problem
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Table 12 A set of trained weights of ANN-TS for each case of scenario 2 of Troesch’s problem
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Raja, M.A.Z., Shah, F.H., Tariq, M. et al. Design of artificial neural network models optimized with sequential quadratic programming to study the dynamics of nonlinear Troesch’s problem arising in plasma physics. Neural Comput & Applic 29, 83–109 (2018). https://doi.org/10.1007/s00521-016-2530-2

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