mlsanityData to 7 Oct 2026

Learning Stable Koopman Embeddings

arXiv 2110.06509 ↗First posted 13 Oct 2021cs.LG cs.SY eess.SYcited by 18

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Abstract (arXiv)

In this paper, we present a new data-driven method for learning stable models of nonlinear systems. Our model lifts the original state space to a higher-dimensional linear manifold using Koopman embeddings. Interestingly, we prove that every discrete-time nonlinear contracting model can be learnt in our framework. Another significant merit of the proposed approach is that it allows for unconstrained optimization over the Koopman embedding and operator jointly while enforcing stability of the model, via a direct parameterization of stable linear systems, greatly simplifying the computations involved. We validate our method on a simulated system and analyze the advantages of our parameterization compared to alternatives.

Key facts. Cited by 18 papers in the map (data through 7 Oct 2026). Builds on 14 of its 35 references that are in the map. First posted on arXiv 13 Oct 2021.

Builds on

14 of 35 references.

References · 14

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 491
  3. 2017Learning Koopman Invariant Subspaces for Dynamic Mode Decompositioncited by 156
  4. 2017Linearly-Recurrent Autoencoder Networks for Learning Dynamicscited by 142
  5. 2017VAMPnets: Deep learning of molecular kineticscited by 135
  6. 2016Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive controlcited by 134
  7. 2017Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systemscited by 125
  8. 2019Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stabilitycited by 53
  9. 2017Koopman-based lifting techniques for nonlinear systems identificationcited by 18
  10. 2017Convex Parameterizations and Fidelity Bounds for Nonlinear Identification and Reduced-Order Modellingcited by 8
  11. 2021On the equivalence of contraction and Koopman approaches for nonlinear stability and controlcited by 6
  12. 2020Luenberger observers for discrete-time nonlinear systemscited by 5
2 more
  1. 2018A note on approximating the nearest stable discrete-time descriptor system with fixed rankcited by 3
  2. 2021Contraction-Based Methods for Stable Identification and Robust Machine Learning: a Tutorialcited by 3

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Cited by year

Citing papers per year: 2021: 2; 2022: 5; 2023: 6; 2024: 3; 2025: 2.

18 papers.

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  1. 2022Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spacescited by 66
  2. 2022Koopman Neural Forecaster for Time Series with Temporal Distribution Shiftscited by 35
  3. 2023Koopman Kernel Regressioncited by 22
  4. 2023Learning invariant representations of time-homogeneous stochastic dynamical systemscited by 17
  5. 2021Diffeomorphically Learning Stable Koopman Operatorscited by 13
  6. 2022Neural Koopman Lyapunov Controlcited by 11
  7. 2024Learning Neural Contracting Dynamics: Extended Linearization and Global Guaranteescited by 11
  8. 2023Efficient Dynamics Modeling in Interactive Environments with Koopman Theorycited by 10
  9. 2023Consistent Long-Term Forecasting of Ergodic Dynamical Systemscited by 10
  10. 2023Neural Koopman prior for data assimilationcited by 9
  11. 2024Learning the Infinitesimal Generator of Stochastic Diffusion Processescited by 9
  12. 2024Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroupscited by 7
6 more
  1. 2025Predictability Enables Parallelization of Nonlinear State Space Modelscited by 7
  2. 2025Augmented Invertible Koopman Autoencoder for long-term time series forecastingcited by 3
  3. 2022Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decompositioncited by 2
  4. 2021Data-driven modelling of nonlinear dynamics by barycentric coordinates and memorycited by 1
  5. 2023Data-Driven Encoding: A New Numerical Method for Computation of the Koopman Operatorcited by 1
  6. 2022Learned Lifted Linearization Applied to Unstable Dynamic Systems Enabled by Koopman Direct Encodingcited by 0