Linearly-Recurrent Autoencoder Networks for Learning Dynamics

TL;DR

This paper introduces a neural network architecture combining autoencoders and linear recurrent dynamics to learn low-dimensional models of nonlinear dynamical systems using Koopman operator theory.

Contribution

It proposes a novel neural network design for learning Koopman-invariant subspaces and offers a method for balanced model reduction in feature space.

Findings

Successfully identified Koopman eigenfunctions of the Duffing equation

Created accurate low-dimensional models of unstable cylinder wake flow

Made short-term predictions of the chaotic Kuramoto-Sivashinsky equation

Abstract

This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural-network approximations of the underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD) provides a useful data-driven approximation of the Koopman operator for analyzing dynamical systems. This paper addresses a fundamental problem associated with EDMD: a trade-off between representational capacity of the dictionary and over-fitting due to insufficient data. A new neural network architecture combining an autoencoder with linear recurrent dynamics in the encoded state is used to learn a low-dimensional and highly informative Koopman-invariant subspace of observables. A method is also presented for balanced model reduction of over-specified EDMD systems in feature space. Nonlinear reconstruction using partially linear multi-kernel regression aims to improve reconstruction accuracy from the low-dimensional state when the data has complex but intrinsically low-dimensional structure. The techniques demonstrate the ability to identify Koopman eigenfunctions of the unforced Duffing equation, create accurate low-dimensional models of an unstable cylinder wake flow, and make short-time predictions of the chaotic Kuramoto-Sivashinsky equation.