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We consider Koopman operator theory in the context of nonlinearinfinite-dimensional systems, where the operator is defined over a space ofbounded continuous functionals. The properties of the Koopman semigroup aredescribed and a finite-dimensional projection of the semigroup is proposed,which provides a linear finite-dimensional approximation of the underlyinginfinite-dimensional dynamics. This approximation is used to obtain spectralproperties from data, a method which can be seen as a generalization ofExtended Dynamic Mode Decomposition for infinite-dimensional systems. Finally,we exploit the proposed framework to identify (a finite-dimensionalapproximation of) the Lie generator associated with the Koopman semigroup. Thisapproach yields a linear method for nonlinear PDE identification, which iscomplemented with theoretical convergence results.

Koopman operator framework for spectral analysis and identification of infinite-dimensional systems