Nonlinear reduction of high-dimensional dynamical systems via neural networks

A technique for empirically determining optimal coordinates for modeling a dynamical system is presented. The methodology may be viewed as a nonlinear extension of the Karhunen-Lo\`eve procedure and is implemented via an autoassociative neural network. Given a high-dimensional system of differential equations which model a dynamical system asymptotically residing on a stable attractor, the task of the network is to compute a reembedding of the attractor and the dynamics into an ambient space which reflects the intrinsic dimensionality of the problem. The method is demonstrated on the unforced Van der Pol oscillator, the forced Van der Pol, and the Kuramoto-Sivashinsky equation.