A hybrid neural network model for the dynamics of the Kuramoto-Sivashinsky equation (Q2566901)

The author uses a hybrid approach consisting of two neural networks to model the oscillatory dynamical behavior of the Kuramoto-Sivashinsky equation at a bifurcation parameter \(\alpha=84.25\). This oscillatory behavior results from a fixed point that occurs at \(\alpha=72\) having a shape of two-humped curve that becomes unstable and undergoes a Hopf bifurcation at \(\alpha=83.75\). First, Karhunen-Löve 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 whose dynamics is similar to the original dynamics of the Kuramoto-Sivashinsky equation was derived via Karhunen-Löve 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. The author shows that a reduced dynamical model of the Kuramoto-Sivashinsky equation is obtained by combining Karhunen-Löve decomposition and neural networks.