Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

TL;DR

This paper introduces gEDMD, a data-driven method extending EDMD for approximating the Koopman generator, applicable to deterministic and stochastic systems, enabling system identification, model reduction, and control.

Contribution

The paper presents gEDMD, a novel extension of EDMD, for approximating the Koopman generator in both deterministic and stochastic systems, facilitating eigenanalysis, system identification, and control.

Findings

Effective approximation of Koopman generator demonstrated on molecular dynamics.

gEDMD accurately identifies drift and diffusion in stochastic systems.

Method enables coarse-grained modeling and control strategies.

Abstract

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.