# Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems

arXiv: https://arxiv.org/abs/2412.02807 · HTML: https://mlsanity.com/paper/2412.02807/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 32 references that are in the map.
- First posted on arXiv 3 Dec 2024.

## Builds on (most cited first, 13 of 13)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees](https://mlsanity.com/paper/2206.01913/) (2022)
- [Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification](https://mlsanity.com/paper/2312.09131/) (2023)
- [Koopman-based lifting techniques for nonlinear systems identification](https://mlsanity.com/paper/1709.02003/) (2017)
- [LyZNet: A Lightweight Python Tool for Learning and Verifying Neural Lyapunov Functions and Regions of Attraction](https://mlsanity.com/paper/2403.10013/) (2024)
- [Koopman-Based Neural Lyapunov Functions for General Attractors](https://mlsanity.com/paper/2203.12303/) (2022)
- [On the equivalence of contraction and Koopman approaches for nonlinear stability and control](https://mlsanity.com/paper/2103.15033/) (2021)
- [Spectral Koopman Method for Identifying Stability Boundary](https://mlsanity.com/paper/2312.06885/) (2023)
- [Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems](https://mlsanity.com/paper/2009.11544/) (2020)
- [Stochastic Reinforcement Learning with Stability Guarantees for Control of Unknown Nonlinear Systems](https://mlsanity.com/paper/2409.08382/) (2024)

## Built on by (most cited first, 1 of 1)

- [Physics-Informed Machine Learning for Characterizing System Stability](https://mlsanity.com/paper/2511.08831/) (2025)

## Abstract (arXiv)

Koopman operator theory has gained significant attention in recent years for identifying discrete-time nonlinear systems by embedding them into an infinite-dimensional linear vector space. However, providing stability guarantees while learning the continuous-time dynamics, especially under conditions of relatively low observation frequency, remains a challenge within the existing Koopman-based learning frameworks. To address this challenge, we propose an algorithmic framework to simultaneously learn the vector field and Lyapunov functions for unknown nonlinear systems, using a limited amount of data sampled across the state space and along the trajectories at a relatively low sampling frequency. The proposed framework builds upon recently developed high-accuracy Koopman generator learning for capturing transient system transitions and physics-informed neural networks for training Lyapunov functions. We show that the learned Lyapunov functions can be formally verified using a satisfiability modulo theories (SMT) solver and provide less conservative estimates of the region of attraction compared to existing methods.