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Numerical solution for high order differential equations using a hybrid neural network-optimization method. (English) Zbl 1105.65340
Summary: This paper reports a novel hybrid method based on optimization techniques and neural networks methods for the solution of high order ordinary differential equations. Here neural networks is considered as a part of large field called neural computing or soft computing. This means that we propose a new solution method for the approximated solution of high order ordinary differential equations using innovative mathematical tools and neural-like systems of computation. This hybrid method can result in improved numerical methods for solving initial/boundary value problems, without using preassigned discretisation points. The mixture of feed forward neural networks and optimization techniques, based on Nelder-Mead method is used to introduce the close analytic form of the solution for the differential equation. Excellent test results are obtained for the solution of lower and higher order differential equations. The model finds approximation solution for the differential equation inside and outside the domain of consideration for the close enough neighborhood of initial/boundary points. Numerical examples are described to demonstrate the method.

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI
[1] Stoer, J.; Bulirsch, R., Introduction to numerical analysis, (1993), Springer-Verlag New York · Zbl 0771.65002
[2] Lambert, J.D., Computational methods in ordinary differential equations, (1983), John Wiley & Sons New York · Zbl 0258.65069
[3] Gottlieb, D.; Orszag, S.A., Numerical analysis of spectral methods: theory and applications, () · Zbl 0561.76076
[4] Lee, H.; Kang, I.S., Neural algorithms for solving differential equations, Journal of computational physics, 91, 110-131, (1990) · Zbl 0717.65062
[5] Meade, A.J.; Fernandez, A.A., The numerical solution of linear ordinary differential equations by feedforward neural networks, Mathematical and computer modelling, 19, 12, 1-25, (1994) · Zbl 0807.65079
[6] Meade, A.J.; Fernandez, A.A., Solution of nonlinear ordinary differential equations by feedforward neural networks, Mathematical and computer modelling, 20, 9, 19-44, (1994) · Zbl 0818.65077
[7] Dissanayake, M.W.M.G.; Phan-Thien, N., Neural-network-based approximations for solving partial differential equations, Communications in numerical methods in engineering, 10, 195-201, (1994) · Zbl 0802.65102
[8] Bo-An Liu, B. Jammes, Solving ordinary differential equations by neural networks, in: Proceeding of 13th European Simulation Multi-Conference Modelling and Simulation: A Tool for the Next Millennium, Warsaw, Poland, June 1-4, 1999.
[9] Lagaris, I.E.; Likas, A.; Fotiadis, D.I., Artificial neural networks for solving ordinary and partial differential equations, IEEE transactions on neural networks, 9, 5, 987-1000, (1998)
[10] K. Valasoulis, D.I. Fotiadis, I.E. Lagaris, A. Likas, Solving differential equations with neural networks: implementation on a DSP platform, in: Proceeding of 14th International Conference on Digital Signal Processing, Santorini, Greece, July 2002, pp. 1265-1268.
[11] Schalkoff, R.J., Artificial neural networks, (1997), McGraw-Hill New York · Zbl 0910.68165
[12] Picton, P., Neural networks, (2000), Palgrave Great Britain
[13] Minsky, M.; Papert, S., Perceptrons, (1969), MIT Press · Zbl 0197.43702
[14] Haykin, S., Neural networks: A comprehensive foundation, (1999), Prentice Hall New Jersey · Zbl 0934.68076
[15] Khanna, T., Foundations of neural networks, (1990), Addison-Wesley Reading, MA · Zbl 0752.92003
[16] Stanley, J., Introduction to neural networks, (1990), Sierra Mardre
[17] Lippmann, R.P., An introduction to computing with neural nets, IEEE ASSP magazine, 4-22, (1987)
[18] Hornick, K.; Stinchcombe, M.; white, H., Multilayer feedforward networks are universal approximators, Neural networks, 2, 5, 359-366, (1989) · Zbl 1383.92015
[19] Lapedes, A.; Farber, R., How neural nets work?, (), 442-456
[20] Mckeown, J.J.; Stella, F.; Hall, G., Some numerical aspects of the training problem for feed-forward neural nets, Neural networks, 10, 8, 1455-1463, (1997)
[21] Kincaid, D.R.; Cheney, E.W., Numerical analysis: mathematics of scientific computing, (2002), Brooks/Cole Pacific Grove, CA
[22] Likas, A.; Karras, D.A.; Lagaris, I.E., Neural-network training and simulation using a multidimensional optimization system, International journal of computer mathematics, 67, 33-46, (1998) · Zbl 0892.68086
[23] Van der Smagt, P., Minimization methods for training feedforward neural networks, Neural networks, 7, 1, 1-11, (1994)
[24] Rao, S.S., Optimization: theory and applications, (1989), Wiley Eastern Limited New Delhi · Zbl 0448.90033
[25] Nelder, J.A.; Mead, R., A simplex method for function minimization, Computer journal, 7, 308-313, (1965) · Zbl 0229.65053
[26] Lagarias, J.C.; Reeds, J.A.; Wright, M.H.; Wright, P.E., Convergence properties of the nelder – mead simplex method in low dimensions, SIAM journal of optimization, 9, 1, 112-147, (1998) · Zbl 1005.90056
[27] Malek, A.; Phillips, T.N., Pseudospectral collocation methods for fourth-order differential equations, IMA journal of numerical analysis, 15, 523-553, (1995) · Zbl 0855.65088
[28] Bernardi, C.; Maday, Y., Some spectral approximations of one-dimensional fourth-order problems, () · Zbl 1004.65119
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