Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
This work provides a data-driven tool for analyzing and controlling complex dynamical systems, but it is incremental as it builds directly on existing methods like EDMD and SINDy.
The authors tackled the problem of approximating the Koopman generator from data, developing a method called gEDMD that extends EDMD for deterministic and stochastic systems, enabling tasks like system identification, model reduction, and control, with efficacy demonstrated on examples including molecular dynamics.
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.