# Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds

arXiv: https://arxiv.org/abs/2603.15091 · HTML: https://mlsanity.com/paper/2603.15091/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 97 references that are in the map.
- First posted on arXiv 16 Mar 2026.

## Builds on (most cited first, 26 of 26)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [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)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Bagging, optimized dynamic mode decomposition (BOP-DMD) for robust, stable forecasting with spatial and temporal uncertainty-quantification](https://mlsanity.com/paper/2107.10878/) (2021)
- [Finite-data error bounds for Koopman-based prediction and control](https://mlsanity.com/paper/2108.07102/) (2021)
- [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022)
- [Kernel methods for detecting coherent structures in dynamical data](https://mlsanity.com/paper/1904.07752/) (2019)
- [Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems](https://mlsanity.com/paper/2308.10697/) (2023)
- [An overview of Koopman-based control: From error bounds to closed-loop guarantees](https://mlsanity.com/paper/2509.02839/) (2025)
- [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)
- [Optimizing Oblique Projections for Nonlinear Systems using Trajectories](https://mlsanity.com/paper/2106.01211/) (2021)
- [Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance](https://mlsanity.com/paper/2207.14387/) (2022)
- [Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2506.15782/) (2025)
- [An Introductory Guide to Koopman Learning](https://mlsanity.com/paper/2510.22002/) (2025)
- [Invariance Proximity: Closed-Form Error Bounds for Finite-Dimensional Koopman-Based Models](https://mlsanity.com/paper/2311.13033/) (2023)

## Built on by (most cited first, 4 of 4)

- [Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors](https://mlsanity.com/paper/2604.01459/) (2026)
- [Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning](https://mlsanity.com/paper/2606.05131/) (2026)
- [Prediction Limits and Koopman Closure of Geometry-Induced Soft State Abstractions](https://mlsanity.com/paper/2609.32652/) (2026)
- [Understanding Latent-Dimension Scaling in Dynamical-System Learning through Spectral Reliability](https://mlsanity.com/paper/2610.11866/) (2026)

## Abstract (arXiv)

Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature spaces induced by a user-chosen dictionary are rarely invariant, so closure failures and projection errors lead to spurious eigenvalues, misleading Koopman modes, and overconfident forecasts. This paper addresses a central validation problem in data-driven Koopman methods: how to quantify invariance and projection errors for an arbitrary feature space using only snapshot data, and how to use these diagnostics to produce actionable guarantees and guide dictionary refinement? A unified a posteriori methodology is developed for certifying when a Koopman approximation is trustworthy and improving it when it is not. Koopman invariance is quantified using principal angles between a subspace and its Koopman image, yielding principal observables and a principal angle decomposition (PAD), a dynamics-informed alternative to SVD truncation with significantly improved performance. Multi-step error bounds are derived for Koopman and Perron--Frobenius mode decompositions, including RKHS-based pointwise guarantees, and are complemented by Gaussian process expected error surrogates. The resulting toolbox enables validated spectral analysis, certified forecasting, and principled dictionary and kernel learning, demonstrated on chaotic and high-dimensional benchmarks and real-world datasets, including cavity flow and the Pluto--Charon system.