mlsanityData to 7 Oct 2026

Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds

arXiv 2603.15091 ↗First posted 16 Mar 2026math.NA cs.LG cs.NAcited by 3

Not on the map. Why?

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.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 26 of its 97 references that are in the map. First posted on arXiv 16 Mar 2026.

Builds on

26 of 97 references.

References · 26

  1. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 491
  2. 2023Resurrecting Recurrent Neural Networks for Long Sequencescited by 397
  3. 2021Modern Koopman Theory for Dynamical Systemscited by 269
  4. 2017Learning Koopman Invariant Subspaces for Dynamic Mode Decompositioncited by 156
  5. 2017Linearly-Recurrent Autoencoder Networks for Learning Dynamicscited by 142
  6. 2017VAMPnets: Deep learning of molecular kineticscited by 135
  7. 2021Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and controlcited by 134
  8. 2017Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systemscited by 125
  9. 2017On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operatorcited by 70
  10. 2017Variational approach for learning Markov processes from time series datacited by 51
  11. 2014Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decompositioncited by 49
  12. 2021Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systemscited by 44
14 more
  1. 2020Kernel-based approximation of the Koopman generator and Schr"odinger operatorcited by 31
  2. 2021Bagging, optimized dynamic mode decomposition (BOP-DMD) for robust, stable forecasting with spatial and temporal uncertainty-quantificationcited by 30
  3. 2021Finite-data error bounds for Koopman-based prediction and controlcited by 26
  4. 2022The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systemscited by 21
  5. 2019Kernel methods for detecting coherent structures in dynamical datacited by 13
  6. 2023Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systemscited by 11
  7. 2025An overview of Koopman-based control: From error bounds to closed-loop guaranteescited by 9
  8. 2024Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Sizecited by 7
  9. 2024Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operatorscited by 7
  10. 2021Optimizing Oblique Projections for Nonlinear Systems using Trajectoriescited by 6
  11. 2022Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariancecited by 6
  12. 2025Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spacescited by 5
  13. 2025An Introductory Guide to Koopman Learningcited by 5
  14. 2023Invariance Proximity: Closed-Form Error Bounds for Finite-Dimensional Koopman-Based Modelscited by 4

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 3.

3 papers.

  1. 2026Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectorscited by 0
  2. 2026Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learningcited by 0
  3. 2026Understanding Latent-Dimension Scaling in Dynamical-System Learning through Spectral Reliabilitycited by 0