# Modern Koopman Theory for Dynamical Systems

arXiv: https://arxiv.org/abs/2102.12086 · HTML: https://mlsanity.com/paper/2102.12086/

## Key facts

- Cited by 276 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 215 references that are in the map.
- First posted on arXiv 24 Feb 2021.

## Builds on (most cited first, 30 of 33)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Variational approach for learning Markov processes from time series data](https://mlsanity.com/paper/1707.04659/) (2017)
- [Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition](https://mlsanity.com/paper/1409.5496/) (2014)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms](https://mlsanity.com/paper/2002.09726/) (2020)
- [Koopman operator-based model reduction for switched-system control of PDEs](https://mlsanity.com/paper/1710.06759/) (2017)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Sparsity-promoting dynamic mode decomposition](https://mlsanity.com/paper/1309.4165/) (2013)
- [Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations](https://mlsanity.com/paper/1610.06773/) (2016)
- [Compressed Dynamic Mode Decomposition for Background Modeling](https://mlsanity.com/paper/1512.04205/) (2015)
- [Randomized Matrix Decompositions using R](https://mlsanity.com/paper/1608.02148/) (2016)
- [Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables](https://mlsanity.com/paper/1808.10551/) (2018)
- [Sparsity-promoting algorithms for the discovery of informative Koopman invariant subspaces](https://mlsanity.com/paper/2002.10637/) (2020)
- [On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems](https://mlsanity.com/paper/1803.08562/) (2018)
- [An Incremental Approach to Online Dynamic Mode Decomposition for Time-Varying Systems with Applications to EEG Data Modeling](https://mlsanity.com/paper/1908.01047/) (2019)
- [Geometric Considerations of a Good Dictionary for Koopman Analysis of Dynamical Systems: Cardinality, 'Primary Eigenfunction,' and Efficient Representation](https://mlsanity.com/paper/1912.09570/) (2019)
- [On Reduced Input-Output Dynamic Mode Decomposition](https://mlsanity.com/paper/1712.08447/) (2017)
- [Power grid transient stabilization using Koopman model predictive control](https://mlsanity.com/paper/1803.10744/) (2018)
- [Tensor-based computation of metastable and coherent sets](https://mlsanity.com/paper/1908.04741/) (2019)

## Built on by (most cited first, 30 of 276)

- [Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors](https://mlsanity.com/paper/2305.18803/) (2023)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Generalized Teacher Forcing for Learning Chaotic Dynamics](https://mlsanity.com/paper/2306.04406/) (2023)
- [Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs](https://mlsanity.com/paper/2310.02619/) (2023)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Learning with 3D rotations, a hitchhiker's guide to SO(3)](https://mlsanity.com/paper/2404.11735/) (2024)
- [Deeptime: a Python library for machine learning dynamical models from time series data](https://mlsanity.com/paper/2110.15013/) (2021)
- [Koopman Neural Forecaster for Time Series with Temporal Distribution Shifts](https://mlsanity.com/paper/2210.03675/) (2022)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [Koopman neural operator as a mesh-free solver of non-linear partial differential equations](https://mlsanity.com/paper/2301.10022/) (2023)
- [Zero-shot forecasting of chaotic systems](https://mlsanity.com/paper/2409.15771/) (2024)
- [Generative Latent Neural PDE Solver using Flow Matching](https://mlsanity.com/paper/2503.22600/) (2025)
- [Koopman Q-learning: Offline Reinforcement Learning via Symmetries of Dynamics](https://mlsanity.com/paper/2111.01365/) (2021)
- [Long-Term Fairness with Unknown Dynamics](https://mlsanity.com/paper/2304.09362/) (2023)
- [Differentiability in Unrolled Training of Neural Physics Simulators on Transient Dynamics](https://mlsanity.com/paper/2402.12971/) (2024)
- [Discovering Sparse Interpretable Dynamics from Partial Observations](https://mlsanity.com/paper/2107.10879/) (2021)
- [Deep Koopman Operator with Control for Nonlinear Systems](https://mlsanity.com/paper/2202.08004/) (2022)
- [Multifactor Sequential Disentanglement via Structured Koopman Autoencoders](https://mlsanity.com/paper/2303.17264/) (2023)
- [Beyond Geometry: Comparing the Temporal Structure of Computation in Neural Circuits with Dynamical Similarity Analysis](https://mlsanity.com/paper/2306.10168/) (2023)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Generative learning for nonlinear dynamics](https://mlsanity.com/paper/2311.04128/) (2023)
- [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022)
- [Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2303.17078/) (2023)
- [Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2501.13329/) (2025)
- [True Zero-Shot Inference of Dynamical Systems Preserving Long-Term Statistics](https://mlsanity.com/paper/2505.13192/) (2025)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Almost-Linear RNNs Yield Highly Interpretable Symbolic Codes in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2410.14240/) (2024)

## Abstract (arXiv)

The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator theoretic or probabilistic frameworks. Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinite-dimensional linear operator acting on the space of all possible measurement functions of the system. This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems. However, obtaining finite-dimensional coordinate systems and embeddings in which the dynamics appear approximately linear remains a central open challenge. The success of Koopman analysis is due primarily to three key factors: 1) there exists rigorous theory connecting it to classical geometric approaches for dynamical systems, 2) the approach is formulated in terms of measurements, making it ideal for leveraging big-data and machine learning techniques, and 3) simple, yet powerful numerical algorithms, such as the dynamic mode decomposition (DMD), have been developed and extended to reduce Koopman theory to practice in real-world applications. In this review, we provide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic developments and highlighting these methods with a diverse range of applications. We also discuss key advances and challenges in the rapidly growing field of machine learning that are likely to drive future developments and significantly transform the theoretical landscape of dynamical systems.