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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.