Modelling the dynamics of nonlinear partial differential equations using neural networks

摘要

The dynamics of two nonlinear partial differential equations (PDEs) known as the Kuramoto–Sivashinsky (K–S) equation and the two-dimensional Navier–Stokes (N–S) equations are analyzed using Karhunen–Loéve (K–L) decomposition and artificial neural networks (ANN). For the K–S equation, numerical simulations using a pseudospectral Galerkin method is presented at a bifurcation parameter α=17.75, where a dynamical behavior represented by a heteroclinic connection is obtained. We apply K–L decomposition on the numerical simulation data with the task of reducing the data into a set of data coefficients. Then we use ANN to model, and predict the data coefficients at a future time. It is found that training the neural networks with only the first data coefficient is enough to capture the underlying dynamics, and to predict for the other remaining data coefficients. As for the two-dimensional N–S equation, a quasiperiodic behavior represented in phase space by a torus is analyzed at Re=14.0. Applying the symmetry observed in the two-dimensional N–S equations on the quasiperiodic behavior, eight different tori were obtained. We show that by exploiting the symmetries of the equation and using K–L decomposition in conjunction with neural networks, a smart neural model is obtained.