  We derive a data-driven method for the approximation of the Koopman generator
called gEDMD, which can be regarded as a straightforward extension of EDMD
(extended dynamic mode decomposition). This approach is applicable to
deterministic and stochastic dynamical systems. It can be used for computing
eigenvalues, eigenfunctions, and modes of the generator and for system
identification. In addition to learning the governing equations of
deterministic systems, which then reduces to SINDy (sparse identification of
nonlinear dynamics), it is possible to identify the drift and diffusion terms
of stochastic differential equations from data. Moreover, we apply gEDMD to
derive coarse-grained models of high-dimensional systems, and also to determine
efficient model predictive control strategies. We highlight relationships with
other methods and demonstrate the efficacy of the proposed methods using
several guiding examples and prototypical molecular dynamics problems.
