Dynamic Mode Decomposition for Continuous Time Systems with the Liouville Operator

Dynamic Mode Decomposition (DMD) has become synonymous with the Koopman operator, where continuous time dynamics are examined through a discrete time proxy determined by a fixed timestep using Koopman (i.e. composition) operators. Using the newly introduced "occupation kernels," the present manuscript develops an approach to DMD that treats continuous time dynamics directly through the Liouville operator. This manuscript outlines the technical and theoretical differences between Koopman based DMD for discrete time systems and Liouville based DMD for continuous time systems, which includes an examination of these operators over several reproducing kernel Hilbert spaces (RKHSs). A comparison between the obtained Liuoville and Koompan modes are presented for systems relating to fluid dynamics and electroencephalography (EEG). Liouville based DMD methods allow for the incorporation of data that is sampled at high frequencies without producing models with excessively large rank, as would be the case with Koopman based DMD, and Liouville based DMD natively allows for data that does not use fixed time steps. Finally, while both Liouville and Koopman operators are modally unbounded operators, scaled Liouville operators are introduced, which are compact for a large class of dynamics.