A Hybrid Neural Network Model for the Dynamics of the Kuramoto-sivashinsky Equation

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 of five 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 used to 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.