  Identifying coordinate transformations that make strongly nonlinear dynamics
approximately linear is a central challenge in modern dynamical systems. These
transformations have the potential to enable prediction, estimation, and
control of nonlinear systems using standard linear theory. The Koopman operator
has emerged as a leading data-driven embedding, as eigenfunctions of this
operator provide intrinsic coordinates that globally linearize the dynamics.
However, identifying and representing these eigenfunctions has proven to be
mathematically and computationally challenging. This work leverages the power
of deep learning to discover representations of Koopman eigenfunctions from
trajectory data of dynamical systems. Our network is parsimonious and
interpretable by construction, embedding the dynamics on a low-dimensional
manifold that is of the intrinsic rank of the dynamics and parameterized by the
Koopman eigenfunctions. In particular, we identify nonlinear coordinates on
which the dynamics are globally linear using a modified auto-encoder. We also
generalize Koopman representations to include a ubiquitous class of systems
that exhibit continuous spectra, ranging from the simple pendulum to nonlinear
optics and broadband turbulence. Our framework parametrizes the continuous
frequency using an auxiliary network, enabling a compact and efficient
embedding at the intrinsic rank, while connecting our models to half a century
of asymptotics. In this way, we benefit from the power and generality of deep
learning, while retaining the physical interpretability of Koopman embeddings.
