  An outstanding challenge in nonlinear systems theory is identification or
learning of a given nonlinear system's Koopman operator directly from data or
models. Advances in extended dynamic mode decomposition approaches and machine
learning methods have enabled data-driven discovery of Koopman operators, for
both continuous and discrete-time systems. Since Koopman operators are often
infinite-dimensional, they are approximated in practice using
finite-dimensional systems. The fidelity and convergence of a given
finite-dimensional Koopman approximation is a subject of ongoing research. In
this paper we introduce a class of Koopman observable functions that confer an
approximate closure property on their corresponding finite-dimensional
approximations of the Koopman operator. We derive error bounds for the fidelity
of this class of observable functions, as well as identify two key learning
parameters which can be used to tune performance. We illustrate our approach on
two classical nonlinear system models: the Van Der Pol oscillator and the
bistable toggle switch.
