# Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces

arXiv: https://arxiv.org/abs/2205.14027 · HTML: https://mlsanity.com/paper/2205.14027/

## Key facts

- Cited by 67 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 81 references that are in the map.
- First posted on arXiv 27 May 2022.

## Builds on (most cited first, 12 of 12)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Kernel methods through the roof: handling billions of points efficiently](https://mlsanity.com/paper/2006.10350/) (2020)
- [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Learning to Forecast Dynamical Systems from Streaming Data](https://mlsanity.com/paper/2109.09703/) (2021)
- [Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1807.09331/) (2018)
- [Leveraging Low-Rank Relations Between Surrogate Tasks in Structured Prediction](https://mlsanity.com/paper/1903.00667/) (2019)
- [Reduced Rank Multivariate Kernel Ridge Regression](https://mlsanity.com/paper/2005.01559/) (2020)

## Built on by (most cited first, 30 of 67)

- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm](https://mlsanity.com/paper/2312.07186/) (2023)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Learning invariant representations of time-homogeneous stochastic dynamical systems](https://mlsanity.com/paper/2307.09912/) (2023)
- [Estimating Koopman operators with sketching to provably learn large scale dynamical systems](https://mlsanity.com/paper/2306.04520/) (2023)
- [Spectral Representation Learning for Conditional Moment Models](https://mlsanity.com/paper/2210.16525/) (2022)
- [Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms](https://mlsanity.com/paper/2405.14778/) (2024)
- [Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems](https://mlsanity.com/paper/2308.10697/) (2023)
- [Neural Koopman prior for data assimilation](https://mlsanity.com/paper/2309.05317/) (2023)
- [Consistent Long-Term Forecasting of Ergodic Dynamical Systems](https://mlsanity.com/paper/2312.13426/) (2023)
- [Learning the Infinitesimal Generator of Stochastic Diffusion Processes](https://mlsanity.com/paper/2405.12940/) (2024)
- [Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal](https://mlsanity.com/paper/2411.19653/) (2024)
- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [Operator World Models for Reinforcement Learning](https://mlsanity.com/paper/2406.19861/) (2024)
- [Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups](https://mlsanity.com/paper/2410.14477/) (2024)
- [Self-Supervised Evolution Operator Learning for High-Dimensional Dynamical Systems](https://mlsanity.com/paper/2505.18671/) (2025)
- [From Biased to Unbiased Dynamics: An Infinitesimal Generator Approach](https://mlsanity.com/paper/2406.09028/) (2024)
- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)
- [SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems](https://mlsanity.com/paper/2402.03145/) (2024)
- [Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems](https://mlsanity.com/paper/2507.07222/) (2025)
- [Universal Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2508.11990/) (2025)
- [A randomized algorithm to solve reduced rank operator regression](https://mlsanity.com/paper/2312.17348/) (2023)
- [Information Shapes Koopman Representation](https://mlsanity.com/paper/2510.13025/) (2025)
- [Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modelling](https://mlsanity.com/paper/2312.07457/) (2023)
- [Long-Context Linear System Identification](https://mlsanity.com/paper/2410.05690/) (2024)

## Abstract (arXiv)

We study a class of dynamical systems modelled as Markov chains that admit an invariant distribution via the corresponding transfer, or Koopman, operator. While data-driven algorithms to reconstruct such operators are well known, their relationship with statistical learning is largely unexplored. We formalize a framework to learn the Koopman operator from finite data trajectories of the dynamical system. We consider the restriction of this operator to a reproducing kernel Hilbert space and introduce a notion of risk, from which different estimators naturally arise. We link the risk with the estimation of the spectral decomposition of the Koopman operator. These observations motivate a reduced-rank operator regression (RRR) estimator. We derive learning bounds for the proposed estimator, holding both in i.i.d. and non i.i.d. settings, the latter in terms of mixing coefficients. Our results suggest RRR might be beneficial over other widely used estimators as confirmed in numerical experiments both for forecasting and mode decomposition.