  The Koopman operator has emerged as a powerful tool for the analysis of
nonlinear dynamical systems as it provides coordinate transformations to
globally linearize the dynamics. While recent deep learning approaches have
been useful in extracting the Koopman operator from a data-driven perspective,
several challenges remain. In this work, we formalize the problem of learning
the continuous-time Koopman operator with deep neural networks in a
measure-theoretic framework. Our approach induces two types of models:
differential and recurrent form, the choice of which depends on the
availability of the governing equations and data. We then enforce a structural
parameterization that renders the realization of the Koopman operator provably
stable. A new autoencoder architecture is constructed, such that only the
residual of the dynamic mode decomposition is learned. Finally, we employ
mean-field variational inference (MFVI) on the aforementioned framework in a
hierarchical Bayesian setting to quantify uncertainties in the characterization
and prediction of the dynamics of observables. The framework is evaluated on a
simple polynomial system, the Duffing oscillator, and an unstable cylinder wake
flow with noisy measurements.
