# Koopman Operator Dynamical Models: Learning, Analysis and Control

arXiv: https://arxiv.org/abs/2102.02522 · HTML: https://mlsanity.com/paper/2102.02522/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Feb 2021.

## Built on by (most cited first, 30 of 42)

- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Unsupervised Learning of Equivariant Structure from Sequences](https://mlsanity.com/paper/2210.05972/) (2022)
- [Diffeomorphically Learning Stable Koopman Operators](https://mlsanity.com/paper/2112.04085/) (2021)
- [Latent Mamba Operator for Partial Differential Equations](https://mlsanity.com/paper/2505.19105/) (2025)
- [Learning the Koopman Eigendecomposition: A Diffeomorphic Approach](https://mlsanity.com/paper/2110.07786/) (2021)
- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)
- [Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Network](https://mlsanity.com/paper/2507.08749/) (2025)
- [SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems](https://mlsanity.com/paper/2402.03145/) (2024)
- [On the lifting and reconstruction of nonlinear systems with multiple invariant sets](https://mlsanity.com/paper/2304.11860/) (2023)
- [Easy attention: A simple attention mechanism for temporal predictions with transformers](https://mlsanity.com/paper/2308.12874/) (2023)
- [CoNO: Complex Neural Operator for Continous Dynamical Physical Systems](https://mlsanity.com/paper/2406.02597/) (2024)
- [Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om method](https://mlsanity.com/paper/2403.02811/) (2024)
- [Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Engine](https://mlsanity.com/paper/2505.10438/) (2025)
- [GeoMaNO: Geometric Mamba Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2505.12020/) (2025)
- [SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting](https://mlsanity.com/paper/2506.14113/) (2025)
- [From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma](https://mlsanity.com/paper/2508.07494/) (2025)
- [HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs](https://mlsanity.com/paper/2605.00820/) (2026)
- [Towards Data-driven LQR with Koopmanizing Flows](https://mlsanity.com/paper/2201.11640/) (2022)
- [Ensemble forecasts in reproducing kernel Hilbert space family](https://mlsanity.com/paper/2207.14653/) (2022)
- [Joint Problems in Learning Multiple Dynamical Systems](https://mlsanity.com/paper/2311.02181/) (2023)
- [Learning Noise-Robust Stable Koopman Operator for Control with Hankel DMD](https://mlsanity.com/paper/2408.06607/) (2024)
- [Spectrally Informed Learning of Fluid Flows](https://mlsanity.com/paper/2408.14407/) (2024)
- [End-to-end guarantees for indirect data-driven control of bilinear systems with finite stochastic data](https://mlsanity.com/paper/2409.18010/) (2024)
- [Kernel-Based Optimal Control: An Infinitesimal Generator Approach](https://mlsanity.com/paper/2412.01591/) (2024)
- [Data-driven system identification using quadratic embeddings of nonlinear dynamics](https://mlsanity.com/paper/2501.08202/) (2025)
- [Understanding the role of autoencoders for stiff dynamical systems using information theory](https://mlsanity.com/paper/2503.06325/) (2025)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [Data driven modeling for self-similar dynamics](https://mlsanity.com/paper/2310.08282/) (2023)

## Abstract (arXiv)

The Koopman operator allows for handling nonlinear systems through a (globally) linear representation. In general, the operator is infinite-dimensional - necessitating finite approximations - for which there is no overarching framework. Although there are principled ways of learning such finite approximations, they are in many instances overlooked in favor of, often ill-posed and unstructured methods. Also, Koopman operator theory has long-standing connections to known system-theoretic and dynamical system notions that are not universally recognized. Given the former and latter realities, this work aims to bridge the gap between various concepts regarding both theory and tractable realizations. Firstly, we review data-driven representations (both unstructured and structured) for Koopman operator dynamical models, categorizing various existing methodologies and highlighting their differences. Furthermore, we provide concise insight into the paradigm's relation to system-theoretic notions and analyze the prospect of using the paradigm for modeling control systems. Additionally, we outline the current challenges and comment on future perspectives.