Linear versus Nonlinear Dimensionality Reduction of High-Dimensional Dynamical Systems

Abstract

Two techniques for dimensionality reduction of high-dimensional dynamical systems are presented. The first is based on Karhunen--Loève (K-L) analysis and the second on autoassociative neural networks (ANNs). First, we analyze the dynamics of two partial differential equations, namely, the one-dimensional (1-d) Kuramoto--Sivashinsky (K-S) equation and the two-dimensional (2-d) Navier--Stokes (N-S) equations. For the 1-d K-S equation, one particular dynamical behavior, represented by a heteroclinic connection in phase space, is investigated. As for the 2-d N-S equations, a quasi-periodic behavior is examined. Coherent structures of both dynamics were extracted spanning linear subspaces with minimum information loss. Then we obtain systems of ordinary differential equations based on sophisticated (K-L) Galerkin-type approximation capturing the dynamics of the attractors of both regimes residing on linear manifolds. Using the K-L data coefficients as inputs to autoassociative neural networks, we are able to reproduce the dynamics of the attractors which now reside on nonlinear manifolds with fewer dimensions than those previously obtained.