mlsanityData to 7 Oct 2026

Learning the Koopman Eigendecomposition: A Diffeomorphic Approach

arXiv 2110.07786 ↗First posted 15 Oct 2021cs.LG cs.SY eess.SYcited by 9

Not on the map. Why?

Abstract (arXiv)

We present a novel data-driven approach for learning linear representations of a class of stable nonlinear systems using Koopman eigenfunctions. By learning the conjugacy map between a nonlinear system and its Jacobian linearization through a Normalizing Flow one can guarantee the learned function is a diffeomorphism. Using this diffeomorphism, we construct eigenfunctions of the nonlinear system via the spectral equivalence of conjugate systems - allowing the construction of linear predictors for nonlinear systems. The universality of the diffeomorphism learner leads to the universal approximation of the nonlinear system's Koopman eigenfunctions. The developed method is also safe as it guarantees the model is asymptotically stable regardless of the representation accuracy. To our best knowledge, this is the first work to close the gap between the operator, system and learning theories. The efficacy of our approach is shown through simulation examples.

Key facts. Cited by 9 papers in the map (data through 7 Oct 2026). Builds on 10 of its 29 references that are in the map. First posted on arXiv 15 Oct 2021.

Builds on

10 of 29 references.

References · 10

  1. 2015Variational Inference with Normalizing Flowscited by 2,161
  2. 2017Linearly-Recurrent Autoencoder Networks for Learning Dynamicscited by 142
  3. 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
  4. 2020Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximatorscited by 85
  5. 2021Koopman Operator Dynamical Models: Learning, Analysis and Controlcited by 42
  6. 2016Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processescited by 38
  7. 2018Optimal construction of Koopman eigenfunctions for prediction and controlcited by 34
  8. 2019Extended Dynamic Mode Decomposition with Learned Koopman Eigenfunctions for Prediction and Controlcited by 14
  9. 2021On the equivalence of contraction and Koopman approaches for nonlinear stability and controlcited by 6
  10. 2021Inverse Reinforcement Learning: A Control Lyapunov Approachcited by 5

Built on by

Cited by year

Citing papers per year: 2021: 1; 2022: 2; 2023: 3; 2024: 2; 2025: 1.

9 papers.

  1. 2022Universal approximation property of invertible neural networkscited by 30
  2. 2023Koopman Kernel Regressioncited by 22
  3. 2021Diffeomorphically Learning Stable Koopman Operatorscited by 13
  4. 2023On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networkscited by 2
  5. 2023Let's do the time-warp-attend: Learning topological invariants of dynamical systemscited by 2
  6. 2022Towards Data-driven LQR with Koopmanizing Flowscited by 1
  7. 2024Learning Noise-Robust Stable Koopman Operator for Control with Hankel DMDcited by 1
  8. 2024Koopman-based Deep Learning for Nonlinear System Estimationcited by 0
  9. 2025Federated Koopman-Reservoir Learning for Large-Scale Multivariate Time-Series Anomaly Detectioncited by 0