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

## TL;DR

This paper introduces a physics-informed probabilistic deep learning framework for stable, measure-theoretic learning of Koopman operators, enabling accurate, uncertainty-aware modeling of nonlinear dynamical systems.

## Contribution

It formalizes continuous-time Koopman operator learning with neural networks, ensuring stability and incorporating uncertainty quantification via hierarchical Bayesian inference.

## Key findings

- Successfully applied to polynomial, oscillator, and fluid flow systems.
- Achieved stable and interpretable linear embeddings of nonlinear dynamics.
- Quantified uncertainties in dynamic predictions.

## Abstract

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.