Evolutionary Computing Approach for the Solution of Initial value Problems in Ordinary Differential Equations
An evolutionary computing technique for solving initial value problems in Ordinary Differential Equations is proposed in this paper. Neural network is used as a universal approximator while the adaptive parameters of neural networks are optimized by genetic algorithm. The solution is achieved on the continuous grid of time instead of discrete as in other numerical techniques. The comparison is carried out with classical numerical techniques and the solution is found with a uniform accuracy of MSE ≈ 10-9 .
[1] D. Kahaner, C. Moler, S. Nash, "Numerical Methods and Software,"
Prentice-Hall, New Jersey, 1989.
[2] K. Kunz, R. Luebbers, "Finite difference time domain method for
electromagnetic," Boca Raton, CRC Press. 1993.
[3] J.R Dormand, P.J Prince, "A family of embedded Runge-Kutta
formulae," Comp. Appl. Math. J., vol. 6, pp. 19, 1980.
[4] MJ. Jang, CL. Chen, YC. Liu, "On solving the initial-value problems
using the differential transform method," J. Comp. Appl. Math. J., vol.
115, pp. 145-160, 2000.
[5] L.F. Shampine, M. W. Reichelt, "The MATLAB ode suite," SIAM
Scientific Computing J., vol.18, pp. 1-22, 1997.
[6] M.H Li, G.J Wang, "Solving differential equation with constant
coefficient by using radial basis function," Shen Yang Institute of
Chemical Technology J., vol.20, pp. 68-72, 2006.
[7] A.J Meada, A.A Fernandez, "The numerical solution of linear ordinary
differential equation by feedforward neural network," Math Compute
Modeling J., vol.19, pp. 1-25, 1994.
[8] C. Monterola, C. Saloma, "Characterizing the dynamics of constrained
physical systems with unsupervised neural network," Phys Rev E 57,
pp.1247R-1250R, 1998.
[9] B.Ph van Milligen, V.Tribaldos, J.A.Jimenez, "Neural network
differential equation and plasma equilibrium solver," Physical Review
Letters 75, pp. 3594-3597, 1995.
[10] L.Qin, M.Yang, moving mass attitude law based on neural network, in
proc. 6th Int. Conf. machine learning and cybernetics, ICMLC, 5, art.
No. 4370622, 2007, pp.2791-2795.
[11] J.H. Holland, "Adaptation in natural and artificial systems," Ann arbor,
MI, University of Michigan press, 1975.
[12] C.R. Houck, J.A. Joines, M.G. Kay, "A genetic algorithm for function
optimization," A matlab implementation, Technical Report NCSU-IE
TR 95-09, North Carolina State University, Raleigh NC,1995.
[13] Zbigniew Michalewicz, "Genetic algorithms + data structure=
Evolution programs," 2nd ed, New York: Springer-verlag, Berlin, 1994.
[14] R. Hetch-Nielsen, "Kolmogorov-s mapping neural network existence
theorem," 1st 1987 IEEE Int. Conf. Neural networks, San Diego CA, 3,
pp. 11.
[15] K.I. Funahashi, "On the approximate realization of continuous
mappings by neural networks," Neural Networks J., vol. 2.issue 3,
pp.183-192, 1989.
[16] C. Chen, "Degree of approximation by superposition of sigmoidal
function," Analysis in theory & appl J., vol. 9, no 3, pp. 17-28, 1993.
[1] D. Kahaner, C. Moler, S. Nash, "Numerical Methods and Software,"
Prentice-Hall, New Jersey, 1989.
[2] K. Kunz, R. Luebbers, "Finite difference time domain method for
electromagnetic," Boca Raton, CRC Press. 1993.
[3] J.R Dormand, P.J Prince, "A family of embedded Runge-Kutta
formulae," Comp. Appl. Math. J., vol. 6, pp. 19, 1980.
[4] MJ. Jang, CL. Chen, YC. Liu, "On solving the initial-value problems
using the differential transform method," J. Comp. Appl. Math. J., vol.
115, pp. 145-160, 2000.
[5] L.F. Shampine, M. W. Reichelt, "The MATLAB ode suite," SIAM
Scientific Computing J., vol.18, pp. 1-22, 1997.
[6] M.H Li, G.J Wang, "Solving differential equation with constant
coefficient by using radial basis function," Shen Yang Institute of
Chemical Technology J., vol.20, pp. 68-72, 2006.
[7] A.J Meada, A.A Fernandez, "The numerical solution of linear ordinary
differential equation by feedforward neural network," Math Compute
Modeling J., vol.19, pp. 1-25, 1994.
[8] C. Monterola, C. Saloma, "Characterizing the dynamics of constrained
physical systems with unsupervised neural network," Phys Rev E 57,
pp.1247R-1250R, 1998.
[9] B.Ph van Milligen, V.Tribaldos, J.A.Jimenez, "Neural network
differential equation and plasma equilibrium solver," Physical Review
Letters 75, pp. 3594-3597, 1995.
[10] L.Qin, M.Yang, moving mass attitude law based on neural network, in
proc. 6th Int. Conf. machine learning and cybernetics, ICMLC, 5, art.
No. 4370622, 2007, pp.2791-2795.
[11] J.H. Holland, "Adaptation in natural and artificial systems," Ann arbor,
MI, University of Michigan press, 1975.
[12] C.R. Houck, J.A. Joines, M.G. Kay, "A genetic algorithm for function
optimization," A matlab implementation, Technical Report NCSU-IE
TR 95-09, North Carolina State University, Raleigh NC,1995.
[13] Zbigniew Michalewicz, "Genetic algorithms + data structure=
Evolution programs," 2nd ed, New York: Springer-verlag, Berlin, 1994.
[14] R. Hetch-Nielsen, "Kolmogorov-s mapping neural network existence
theorem," 1st 1987 IEEE Int. Conf. Neural networks, San Diego CA, 3,
pp. 11.
[15] K.I. Funahashi, "On the approximate realization of continuous
mappings by neural networks," Neural Networks J., vol. 2.issue 3,
pp.183-192, 1989.
[16] C. Chen, "Degree of approximation by superposition of sigmoidal
function," Analysis in theory & appl J., vol. 9, no 3, pp. 17-28, 1993.
@article{"International Journal of Electrical, Electronic and Communication Sciences:61875", author = "A. Junaid and M. A. Z. Raja and I. M. Qureshi", title = "Evolutionary Computing Approach for the Solution of Initial value Problems in Ordinary Differential Equations", abstract = "An evolutionary computing technique for solving initial value problems in Ordinary Differential Equations is proposed in this paper. Neural network is used as a universal approximator while the adaptive parameters of neural networks are optimized by genetic algorithm. The solution is achieved on the continuous grid of time instead of discrete as in other numerical techniques. The comparison is carried out with classical numerical techniques and the solution is found with a uniform accuracy of MSE ≈ 10-9 .
", keywords = "Neural networks, Unsupervised learning,
Evolutionary computing, Numerical methods, Fitness evaluation
function.", volume = "3", number = "7", pages = "1485-4", }
