  Koopman analysis provides a general framework from which to analyze a
nonlinear dynamical system in terms of a linear operator acting on an
infinite-dimensional observable space. This theoretical framework provides a
rigorous underpinning for widely used dynamic mode decomposition algorithms.
While such methods have proven to be remarkably useful in the analysis of
time-series data, the resulting linear models must generally be of high order
to accurately approximate fundamentally nonlinear behaviors. This issue poses
an inherent risk of overfitting to training data thereby limiting predictive
capabilities. By contrast, this work explores strategies for nonlinear
data-driven estimation of the action of the Koopman operator. General
strategies that yield nonlinear models are presented for systems both with and
without control. Subsequent projection of the resulting nonlinear equations
onto a low-rank basis yields a low order representation for the underlying
dynamical system. In both computational and experimental examples considered in
this work, linear estimators of the Koopman operator are generally only able to
provide short-term predictions for the observable dynamics while comparable
nonlinear estimators provide accurate predictions on substantially longer
timescales and replicate infinite-time behaviors that linear predictors cannot.
