  We introduce two novel generalizations of the Koopman operator method of
nonlinear dynamic modeling. Each of these generalizations leads to greatly
improved predictive performance without sacrificing a unique trait of Koopman
methods: the potential for fast, globally optimal control of nonlinear,
nonconvex systems. The first generalization, Convex Koopman Models, uses convex
rather than linear dynamics in the lifted space. The second, Extended Koopman
Models, additionally introduces an invertible transformation of the control
signal which contributes to the lifted convex dynamics. We describe a deep
learning architecture for parameterizing these classes of models, and show
experimentally that each significantly outperforms traditional Koopman models
in trajectory prediction for two nonlinear, nonconvex dynamic systems.
