mlsanityData to 14 Oct 2026

Linearly-Recurrent Autoencoder Networks for Learning Dynamics

arXiv 1712.01378 ↗First posted 4 Dec 2017math.DS cs.LG stat.MLcited by 145

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Abstract (arXiv)

This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural-network approximations of the underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD) provides a useful data-driven approximation of the Koopman operator for analyzing dynamical systems. This paper addresses a fundamental problem associated with EDMD: a trade-off between representational capacity of the dictionary and over-fitting due to insufficient data. A new neural network architecture combining an autoencoder with linear recurrent dynamics in the encoded state is used to learn a low-dimensional and highly informative Koopman-invariant subspace of observables. A method is also presented for balanced model reduction of over-specified EDMD systems in feature space. Nonlinear reconstruction using partially linear multi-kernel regression aims to improve reconstruction accuracy from the low-dimensional state when the data has complex but intrinsically low-dimensional structure. The techniques demonstrate the ability to identify Koopman eigenfunctions of the unforced Duffing equation, create accurate low-dimensional models of an unstable cylinder wake flow, and make short-time predictions of the chaotic Kuramoto-Sivashinsky equation.

Key facts. Cited by 145 papers in the map (data through 14 Oct 2026). First posted on arXiv 4 Dec 2017.

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Citing papers per year: 2021: 24; 2022: 18; 2023: 26; 2024: 22; 2025: 24; 2026: 31.

145 papers.

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  1. 2021Enhancing Computational Fluid Dynamics with Machine Learningcited by 138
  2. 2023Machine learning with data assimilation and uncertainty quantification for dynamical systems: a reviewcited by 75
  3. 2022Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoderscited by 58
  4. 2023Generalized Teacher Forcing for Learning Chaotic Dynamicscited by 58
  5. 2021Data-driven discovery of intrinsic dynamicscited by 45
  6. 2021Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systemscited by 45
  7. 2021Deeptime: a Python library for machine learning dynamical models from time series datacited by 38
  8. 2022Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal datacited by 37
  9. 2023Deep networks for system identification: a Surveycited by 37
  10. 2024Out-of-Domain Generalization in Dynamical Systems Reconstructioncited by 33
  11. 2023Koopman neural operator as a mesh-free solver of non-linear partial differential equationscited by 29
  12. 2022Learning continuous models for continuous physicscited by 28
38 more
  1. 2022A Comprehensive Review of Digital Twin -- Part 1: Modeling and Twinning Enabling Technologiescited by 28
  2. 2022Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifoldscited by 25
  3. 2023Multifactor Sequential Disentanglement via Structured Koopman Autoencoderscited by 24
  4. 2023Koopman Kernel Regressioncited by 22
  5. 2023Machine Learning for Partial Differential Equationscited by 21
  6. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  7. 2025Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networkscited by 20
  8. 2025True Zero-Shot Inference of Dynamical Systems Preserving Long-Term Statisticscited by 20
  9. 2024Almost-Linear RNNs Yield Highly Interpretable Symbolic Codes in Dynamical Systems Reconstructioncited by 19
  10. 2021Learning Stable Koopman Embeddingscited by 18
  11. 2023Multi-fidelity reduced-order surrogate modelingcited by 18
  12. 2023Learning invariant representations of time-homogeneous stochastic dynamical systemscited by 17
  13. 2024Temporally Consistent Koopman Autoencoders for Forecasting Dynamical Systemscited by 15
  14. 2021Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolutioncited by 14
  15. 2024On latent dynamics learning in nonlinear reduced order modelingcited by 14
  16. 2021Applying Machine Learning to Study Fluid Mechanicscited by 13
  17. 2023Koopman operator learning using invertible neural networkscited by 12
  18. 2023Mori-Zwanzig latent space Koopman closure for nonlinear autoencodercited by 12
  19. 2024Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learningcited by 12
  20. 2021Model reduction for the material point method via an implicit neural representation of the deformation mapcited by 11
  21. 2021Deep Identification of Nonlinear Systems in Koopman Formcited by 11
  22. 2021Nonlinear proper orthogonal decomposition for convection-dominated flowscited by 11
  23. 2021Automated Urban Planning for Reimagining City Configuration via Adversarial Learning: Quantification, Generation, and Evaluationcited by 11
  24. 2023Interpretable learning of effective dynamics for multiscale systemscited by 11
  25. 2024Koopman-Assisted Reinforcement Learningcited by 11
  26. 2023PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operatorcited by 10
  27. 2023Neural Koopman prior for data assimilationcited by 10
  28. 2021Learning the Koopman Eigendecomposition: A Diffeomorphic Approachcited by 9
  29. 2023KoopmanLab: machine learning for solving complex physics equationscited by 9
  30. 2025One-Step Offline Distillation of Diffusion-based Models via Koopman Modelingcited by 9
  31. 2026A Control Theory of Predictability in Latent World Modelscited by 9
  32. 2021Performance and accuracy assessments of an incompressible fluid solver coupled with a deep Convolutional Neural Networkcited by 8
  33. 2021Symmetry-Aware Autoencoders: s-PCA and s-nlPCAcited by 8
  34. 2022CD-ROM: Complemented Deep-Reduced Order Modelcited by 8
  35. 2022Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoderscited by 8
  36. 2021State estimation with limited sensors -- A deep learning based approachcited by 7
  37. 2022Koopman operator for time-dependent reliability analysiscited by 7
  38. 2023Learning to Predict 3D Rotational Dynamics from Images of a Rigid Body with Unknown Mass Distributioncited by 7

50 most cited of 145 All 145 (JSON)