  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.
