# ANN-based modeling of third order runge kutta method

• ## Authors

• M. Dehghanpour
• A. Rahati
• E. Dehghanian
2015-04-12
• Differential Equations, Artificial Neural Network, Runge Kutta Method.
• The world's common rules (Quantum Physics, Electronics, Computational Chemistry and Astronomy) find their normal mathematical explanation in language of differential equations, so finding optimum numerical solution methods for these equations are very important. In this paper, using an artificial neural network (ANN) a numerical approach is designed to solve a specific system of differential equations such that the training process of the ANNÂ  calculates theÂ  optimal values for the coefficients of third order Runge Kutta method. To validate our approach, we performed some experiments by solving two body problem using coefficients obtained by ANN and also two other well-known coefficients namely Classical and Heun. The results show that the ANN approach has a better performance in compare with two other approaches.

• ## References

1. [1] Anastassi A.,"Constructing Rungeâ€“Kutta methods with the use of artificial neural networks", Neural Comput & Applic, Vol. 25, ( 2013), pp.229- 236, available online: http:// arxiv.org/pdf/1106.1194

[2] Burden L, Faires J, Numerical Analisis, Cengage Learning, (2001), pp: 328-335.

[3] Atkinson K, Han W, Stewart D, Numerical solution of ordinary differential equations, John Wiley & Sons, (2009), PA: 15. http://dx.doi.org/10.1002/9781118164495.

[4] Dissanayake M.W.M.G, Phan-Thien N, "Neural-network based approximations for solving partial differential equations", Communicational in Numerical Methods in Engineering, Vol. 25, (1994), PP. 195â€“201.

[5] Meade AJ. Jr, Fernandez A.A, "Solution of nonlinear ordinary differential equations by feedforward neural networks", Mathematical and Computer Modelling, Vol. 20, (1994), pp. 19 â€“ 44, available online: http://www.sciencedirect.com/science/article/pii/089571779400160X. http://dx.doi.org/10.1016/0895-7177(94)00160-X.

[6] Meade AJ.Jr, Fernandez A.A, "The numerical solution of linear ordinary differential equations by feedforward neural networks", Mathematical and Computer Modelling, Vol. 19, No. 12, (1994), pp. 1â€“25, available online: http://www.sciencedirect.com/science/article/pii/0895717794900957. http://dx.doi.org/10.1016/0895-7177(94)90095-7.

[7] Puffer F, Tetzlaff R, Wolf D, "Learning algorithm for cellular neural networks (CNN) solving nonlinear partial differential equations", proceedings international symposium on singnals, systems and Electronics, (1995), pp. 501â€“504.

[8] He S, Reif K, Unbehauen RS, "Multilayer neural networks for solving a class of partial differential equations", Neural Networks, Vol. 13, (2000), pp. 385â€“396, available online: http://www.sciencedirect.com/science/article/pii/S0893608000000137. http://dx.doi.org/10.1109/ISSSE.1995.498041.

[9] Alexandridis A, Chondrodima E, "Sarimveis H, Radial basis function network training using a nonsymmetric partition of the input space and particle swarm optimization", IEEE Transactions on Neural Networks and Learning Sysemst, Vol. 24, No. 2, (2013), pp. 219 â€“230, available online: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6395832.

[10] Jianyu L, Siwei L, Yingjian, Q., Yaping, H., "Numerical solution of differential equations by radial basis function neural networks", Proceedings of International Joint Conference on Neural Networks, (2002),2 pp: 773â€“777, http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1005571.

[11] Ramuhalli P, Udpa L, Udpa S.S, "Finite-element neural networks for solving differential equations", IEEE Transactions on Neural Networks, Vol. 16, (2005), pp. 1381â€“1392, available online: http://people.sc.fsu.edu/~jburkardt/classes/sem_2015/ramuhalli.pdf. http://dx.doi.org/10.1109/TNN.2005.857945.

[12] Butcher J.C, Numerical methods for ordinary differential equations, Wiley, (2003). http://dx.doi.org/10.1002/0470868279.

[13] Haykin S, Neural networks: a comprehensive foundation, Prentice Hall Englewood Cliffs, (1999).

[14] Yu, C.C, Liu, B.D, "A back-propagation algorithm with adaptive learning rate and momentum coefficient", Proceedings of 2002 International Jointt Conference on Neural Networks, (2002) pp: 1218-1223, http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1007668&url

[15] Vahidi J, Qasem Pour S, Numerical calculations Methods, Computer Science, (1969), pp: 247-257.