Modelling the Dynamics of the Kuramoto-Sivashinsky Equation

N. Smaoui (Kuwait)


1-d Kuramoto-Sivashinsky equation, neural networks, modelling.


A hybrid approach consisting of two neural networks is used to model the oscillatory dynamical behavior of the Kuramoto-Sivashinsky (K-S) equation at a bifurcation pa rameter = 84.25. This oscillatory behavior results from a fixed point that occurs at = 72 having a shape of two humped curve that becomes unstable and undergoes a Hopf bifurcation at = 83.75. First, Karhunen-Loeve (K-L) de composition was used to extract five coherent structures of the oscillatory behavior capturing almost 100% of the en ergy. Based on the five coherent structures, a system of five ordinary differential equations (ODEs) whose dynam ics is similar to the original dynamics of the K-S equation was derived via K-L Galerkin projection. Then, an autoas sociative 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 feed-forward neural network was used to model the dy namics at a future time. We show that by combining K-L decomposition and neural networks, a reduced dynamical model of the K-S equation is obtained.

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