Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability
This work addresses the problem of stable and uncertain-aware linear embeddings for nonlinear dynamics, which is incremental as it builds on existing deep learning approaches with added guarantees and probabilistic modeling.
The authors tackled the challenge of learning the Koopman operator for nonlinear dynamical systems by developing a physics-informed probabilistic framework that ensures stability and quantifies uncertainties, achieving accurate predictions on benchmark systems like the Duffing oscillator and unstable cylinder wake flow with noisy data.
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.