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 discretized and examined 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 Koopman and Liouville operators over several reproducing kernel Hilbert spaces. While Liouville operators are modally unbounded, this manuscript introduces the concept of a scaled Liouville operator, which, for many dynamical systems, is a compact operator over the native space of the exponential dot product kernel. Compactness of scaled Liouville operators allows for norm convergence of Liouville-based DMD, which is a decided advantage over Koopman-based DMD.