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Avoiding spectral pollution for transfer operators using residuals

arXiv 2507.16915 ↗First posted 22 Jul 2025math.DS cs.LG cs.NAcited by 0

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Abstract (arXiv)

Koopman operator theory enables linear analysis of nonlinear dynamical systems by lifting their evolution to infinite-dimensional function spaces. However, finite-dimensional approximations of Koopman and transfer (Frobenius--Perron) operators are prone to spectral pollution, introducing spurious eigenvalues that can compromise spectral computations. While recent advances have yielded provably convergent methods for Koopman operators, analogous tools for general transfer operators remain limited. In this paper, we present algorithms for computing spectral properties of transfer operators without spectral pollution, including extensions to the Hardy-Hilbert space. Case studies--ranging from families of Blaschke maps with known spectrum to a molecular dynamics model of protein folding--demonstrate the accuracy and flexibility of our approach. Notably, we demonstrate that spectral features can arise even when the corresponding eigenfunctions lie outside the chosen space, highlighting the functional-analytic subtleties in defining the "true" Koopman spectrum. Our methods offer robust tools for spectral estimation across a broad range of applications.

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References · 17

  1. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 494
  2. 2021Modern Koopman Theory for Dynamical Systemscited by 276
  3. 2017Linearly-Recurrent Autoencoder Networks for Learning Dynamicscited by 145
  4. 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
  5. 2017Variational approach for learning Markov processes from time series datacited by 52
  6. 2022Residual Dynamic Mode Decomposition: Robust and verified Koopmanismcited by 36
  7. 2013Sparsity-promoting dynamic mode decompositioncited by 30
  8. 2022The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systemscited by 21
  9. 2021Deep Learning Enhanced Dynamic Mode Decompositioncited by 19
  10. 2024Adversarial dynamical systems characterize when data-driven learning succeeds or failscited by 10
  11. 2018A kernel-based approach to molecular conformation analysiscited by 8
  12. 2024Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Sizecited by 7
5 more
  1. 2024Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operatorscited by 7
  2. 2025Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spacescited by 6
  3. 2020Transition path theory for Langevin dynamics on manifold: optimal control and data-driven solvercited by 4
  4. 2023Orthogonal polynomial approximation and Extended Dynamic Mode Decomposition in chaoscited by 4
  5. 2024Multiplicative Dynamic Mode Decompositioncited by 4

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