mlsanityData to 7 Oct 2026

Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition

arXiv 1710.04340 ↗First posted 12 Oct 2017cs.LG math.DS stat.MLcited by 156

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Abstract (arXiv)

Spectral decomposition of the Koopman operator is attracting attention as a tool for the analysis of nonlinear dynamical systems. Dynamic mode decomposition is a popular numerical algorithm for Koopman spectral analysis; however, we often need to prepare nonlinear observables manually according to the underlying dynamics, which is not always possible since we may not have any a priori knowledge about them. In this paper, we propose a fully data-driven method for Koopman spectral analysis based on the principle of learning Koopman invariant subspaces from observed data. To this end, we propose minimization of the residual sum of squares of linear least-squares regression to estimate a set of functions that transforms data into a form in which the linear regression fits well. We introduce an implementation with neural networks and evaluate performance empirically using nonlinear dynamical systems and applications.

Key facts. Cited by 156 papers in the map (data through 7 Oct 2026). First posted on arXiv 12 Oct 2017.

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Citing papers per year: 2021: 29; 2022: 26; 2023: 25; 2024: 21; 2025: 33; 2026: 22.

156 papers.

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  1. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
  2. 2021Enhancing Computational Fluid Dynamics with Machine Learningcited by 137
  3. 2023Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictorscited by 110
  4. 2021Chaos as an interpretable benchmark for forecasting and data-driven modellingcited by 87
  5. 2023Machine learning with data assimilation and uncertainty quantification for dynamical systems: a reviewcited by 74
  6. 2022When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learningcited by 64
  7. 2022Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoderscited by 58
  8. 2023Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEscited by 48
  9. 2021Data-driven discovery of intrinsic dynamicscited by 45
  10. 2024Frequency Adaptive Normalization For Non-stationary Time Series Forecastingcited by 45
  11. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 43
  12. 2021Noisy Recurrent Neural Networkscited by 42
38 more
  1. 2021Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modelingcited by 40
  2. 2022Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal datacited by 37
  3. 2021Deeptime: a Python library for machine learning dynamical models from time series datacited by 36
  4. 2022Koopman Neural Forecaster for Time Series with Temporal Distribution Shiftscited by 35
  5. 2023Discovering Causal Relations and Equations from Datacited by 33
  6. 2026Temporal Straightening for Latent Planningcited by 32
  7. 2023Koopman neural operator as a mesh-free solver of non-linear partial differential equationscited by 29
  8. 2021Discovering Sparse Interpretable Dynamics from Partial Observationscited by 25
  9. 2023Multifactor Sequential Disentanglement via Structured Koopman Autoencoderscited by 23
  10. 2023Generative learning for nonlinear dynamicscited by 22
  11. 2021Reduced operator inference for nonlinear partial differential equationscited by 21
  12. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  13. 2023Machine Learning for Partial Differential Equationscited by 20
  14. 2021Deep Learning Enhanced Dynamic Mode Decompositioncited by 19
  15. 2021Learning Stable Koopman Embeddingscited by 18
  16. 2023Estimating Koopman operators with sketching to provably learn large scale dynamical systemscited by 15
  17. 2024Temporally Consistent Koopman Autoencoders for Forecasting Dynamical Systemscited by 15
  18. 2021Applying Machine Learning to Study Fluid Mechanicscited by 13
  19. 2021An Operator Theoretic Approach for Analyzing Sequence Neural Networkscited by 12
  20. 2023Koopman operator learning using invertible neural networkscited by 12
  21. 2023Mori-Zwanzig latent space Koopman closure for nonlinear autoencodercited by 12
  22. 2021Model reduction for the material point method via an implicit neural representation of the deformation mapcited by 11
  23. 2021Deep Identification of Nonlinear Systems in Koopman Formcited by 11
  24. 2024Koopman-Assisted Reinforcement Learningcited by 11
  25. 2024Hamilton-Jacobi Reachability in Reinforcement Learning: A Surveycited by 11
  26. 2021Reproducing kernel Hilbert C*-module and kernel mean embeddingscited by 10
  27. 2022Predicting the temporal dynamics of turbulent channels through deep learningcited by 10
  28. 2023PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operatorcited by 10
  29. 2022Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learningcited by 9
  30. 2023KoopmanLab: machine learning for solving complex physics equationscited by 9
  31. 2023Interpreting and generalizing deep learning in physics-based problems with functional linear modelscited by 9
  32. 2024Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spacescited by 9
  33. 2025One-Step Offline Distillation of Diffusion-based Models via Koopman Modelingcited by 9
  34. 2021Innovations Autoencoder and its Application in One-class Anomalous Sequence Detectioncited by 7
  35. 2021Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraintcited by 7
  36. 2021Data-driven Analysis for Understanding Team Sports Behaviorscited by 6
  37. 2022Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectoriescited by 6
  38. 2023Challenges and opportunities for machine learning in multiscale computational modelingcited by 6

50 most cited of 156 All 156 (JSON)