Koopman Operator Theory for Infinite-Dimensional Systems: Extended Dynamic Mode Decomposition and Identification of Nonlinear PDEs

We consider Koopman operator theory in the context of nonlinear infinitedimensional systems, where the operator is defined over a space of nonlinear functionals. The properties of the Koopman semigroup are described and a finite-dimensional projection of the semigroup is proposed, which provides a linear finite-dimensional approximation of the underlying infinite-dimensional dynamics. This approximation is used to obtain spectral properties from data, a method which can be seen as a generalization of Extended Dynamic Mode Decomposition for infinite-dimensional systems. Finally, we exploit the proposed framework to identify (a finite-dimensional approximation of) the Lie generator associated with the Koopman semigroup. This approach yields a linear method for nonlinear PDE identification, which is complemented with theoretical convergence results.