**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**31106

##### Evolutionary Computing Approach for the Solution of Initial value Problems in Ordinary Differential Equations

**Authors:**
A. Junaid,
M. A. Z. Raja,
I. M. Qureshi

**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,
Numerical Methods,
Evolutionary computing,
unsupervised learning,
Fitness evaluation
function

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1080024

**References:**

[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.