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

arXiv: https://arxiv.org/abs/1906.03663 · HTML: https://mlsanity.com/paper/1906.03663/

## Key facts

- Cited by 53 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jun 2019.

## Built on by (most cited first, 30 of 53)

- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs](https://mlsanity.com/paper/2310.02619/) (2023)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [Noisy Recurrent Neural Networks](https://mlsanity.com/paper/2102.04877/) (2021)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [Koopman neural operator as a mesh-free solver of non-linear partial differential equations](https://mlsanity.com/paper/2301.10022/) (2023)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [AutoIP: A United Framework to Integrate Physics into Gaussian Processes](https://mlsanity.com/paper/2202.12316/) (2022)
- [A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurements](https://mlsanity.com/paper/2204.00205/) (2022)
- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution](https://mlsanity.com/paper/2103.14629/) (2021)
- [Deep Neural Operator Enabled Digital Twin Modeling for Additive Manufacturing](https://mlsanity.com/paper/2405.09572/) (2024)
- [Diffeomorphically Learning Stable Koopman Operators](https://mlsanity.com/paper/2112.04085/) (2021)
- [Physics-aware, probabilistic model order reduction with guaranteed stability](https://mlsanity.com/paper/2101.05834/) (2021)
- [Koopman operator learning using invertible neural networks](https://mlsanity.com/paper/2306.17396/) (2023)
- [Fourier neural operator for real-time simulation of 3D dynamic urban microclimate](https://mlsanity.com/paper/2308.03985/) (2023)
- [Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder](https://mlsanity.com/paper/2310.10745/) (2023)
- [Nonlinear proper orthogonal decomposition for convection-dominated flows](https://mlsanity.com/paper/2110.08295/) (2021)
- [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023)
- [Neural Koopman prior for data assimilation](https://mlsanity.com/paper/2309.05317/) (2023)
- [KoopmanLab: machine learning for solving complex physics equations](https://mlsanity.com/paper/2301.01104/) (2023)
- [CD-ROM: Complemented Deep-Reduced Order Model](https://mlsanity.com/paper/2202.10746/) (2022)
- [Non-equispaced Fourier Neural Solvers for PDEs](https://mlsanity.com/paper/2212.04689/) (2022)
- [The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operators](https://mlsanity.com/paper/2510.06646/) (2025)
- [Invariant preservation in machine learned PDE solvers via error correction](https://mlsanity.com/paper/2303.16110/) (2023)
- [CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilation](https://mlsanity.com/paper/2410.20072/) (2024)
- [Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories](https://mlsanity.com/paper/2209.09977/) (2022)
- [Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Network](https://mlsanity.com/paper/2507.08749/) (2025)

## Abstract (arXiv)

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.