Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability

The Koopman operator has emerged as a powerful tool for the analysis of nonlinear dynamical systems as it provides coordinate transformations which can globally linearize the dynamics. Recent deep learning approaches such as Linearly-Recurrent Autoencoder Networks (LRAN) show great promise for discovering the Koopman operator for a general nonlinear dynamical system from a data-driven perspective, but several challenges remain. In this work, we formalize the problem of learning the continuous-time Koopman operator with deep neural nets in a measure-theoretic framework. This induces two forms of models: differential and recurrent form, the choice of which depends on the availability of the governing equation 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.