# An Introductory Guide to Koopman Learning

arXiv: https://arxiv.org/abs/2510.22002 · HTML: https://mlsanity.com/paper/2510.22002/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 82 references that are in the map.
- First posted on arXiv 24 Oct 2025.

## Builds on (most cited first, 11 of 11)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [Sparsity-promoting dynamic mode decomposition](https://mlsanity.com/paper/1309.4165/) (2013)
- [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022)
- [Adversarial dynamical systems characterize when data-driven learning succeeds or fails](https://mlsanity.com/paper/2407.06312/) (2024)
- [Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Size](https://mlsanity.com/paper/2403.05891/) (2024)
- [Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators](https://mlsanity.com/paper/2405.00782/) (2024)
- [Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2506.15782/) (2025)
- [Orthogonal polynomial approximation and Extended Dynamic Mode Decomposition in chaos](https://mlsanity.com/paper/2305.08074/) (2023)

## Built on by (most cited first, 6 of 6)

- [Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds](https://mlsanity.com/paper/2603.15091/) (2026)
- [Weighted Birkhoff Averages Accelerate Data-Driven Methods](https://mlsanity.com/paper/2511.17772/) (2025)
- [Residual-Guided Dictionary Learning for Spectrally Accurate Koopman Approximation](https://mlsanity.com/paper/2606.29083/) (2026)
- [Learning Ergodic Dynamical Systems from a Finite Trajectory](https://mlsanity.com/paper/2607.22399/) (2026)
- [On the Unique Recovery of Transport Maps and Vector Fields from Finite Measure-Valued Data](https://mlsanity.com/paper/2604.07671/) (2026)
- [Data-driven Koopman mode approximation: A neural power iteration algorithm](https://mlsanity.com/paper/2608.26943/) (2026)

## Abstract (arXiv)

Koopman operators provide a linear framework for data-driven analyses of nonlinear dynamical systems, but their infinite-dimensional nature presents major computational challenges. In this article, we offer an introductory guide to Koopman learning, emphasizing rigorously convergent data-driven methods for forecasting and spectral analysis. We provide a unified account of error control via residuals in both finite- and infinite-dimensional settings, an elementary proof of convergence for generalized Laplace analysis -- a variant of filtered power iteration that works for operators with continuous spectra and no spectral gaps -- and review state-of-the-art approaches for computing continuous spectra and spectral measures. The goal is to provide both newcomers and experts with a clear, structured overview of reliable data-driven techniques for Koopman spectral analysis.