Mathematical Problems in Engineering
Volume 2004 (2004), Issue 3, Pages 305-321

A hybrid neural network model for the dynamics of the Kuramoto-Sivashinsky equation

Nejib Smaoui

Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

Received 2 January 2004; Revised 14 February 2004

Copyright © 2004 Nejib Smaoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A hybrid approach consisting of two neural networks is used to model the oscillatory dynamical behavior of the Kuramoto-Sivashinsky (KS) equation at a bifurcation parameter α=84.25. This oscillatory behavior results from a fixed point that occurs at α=72 having a shape of two-humped curve that becomes unstable and undergoes a Hopf bifurcation at α=83.75. First, Karhunen-Loève (KL) decomposition was used to extract five coherent structures of the oscillatory behavior capturing almost 100% of the energy. Based on the five coherent structures, a system offive ordinary differential equations (ODEs) whose dynamics is similar to the original dynamics of the KS equation was derived via KL Galerkin projection. Then, an autoassociative neural network was utilized on the amplitudes of the ODEs system with the task of reducing the dimension of the dynamical behavior to its intrinsic dimension, and a feedforward neural network was usedto model the dynamics at a future time. We show that by combining KL decomposition and neural networks, a reduced dynamical model of the KS equation is obtained.