mlsanityData to 7 Oct 2026

Kernel-based approximation of the Koopman generator and Schr"odinger operator

arXiv 2005.13231 ↗First posted 27 May 2020math.DS cs.NA math-phcited by 31

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Abstract (arXiv)

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schr"odinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schr"odinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). First posted on arXiv 27 May 2020.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 5; 2022: 2; 2023: 5; 2024: 6; 2025: 5; 2026: 8.

31 papers.

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  1. 2021Modern Koopman Theory for Dynamical Systemscited by 269
  2. 2021Learning to Forecast Dynamical Systems from Streaming Datacited by 17
  3. 2023Learning invariant representations of time-homogeneous stochastic dynamical systemscited by 17
  4. 2022One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernelscited by 15
  5. 2023Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systemscited by 15
  6. 2021Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Seriescited by 14
  7. 2024Dynamical systems and complex networks: A Koopman operator perspectivecited by 9
  8. 2021On the Universal Transformation of Data-Driven Models to Control Systemscited by 5
  9. 2021Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistrycited by 5
  10. 2024Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operatorcited by 5
  11. 2024Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approachcited by 4
  12. 2023Bridging Algorithmic Information Theory and Machine Learning: A New Approach to Kernel Learningcited by 3
19 more
  1. 2026Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Boundscited by 3
  2. 2024Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om methodcited by 2
  3. 2022Ensemble forecasts in reproducing kernel Hilbert space familycited by 1
  4. 2024Nonparametric Sparse Online Learning of the Koopman Operatorcited by 1
  5. 2026Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximationcited by 1
  6. 2026Kernel Methods for Some Transport Equations with Application to Learning Kernels for the Approximation of Koopman Eigenfunctions: A Unified Approach via Variational Methods, Green's Functions and the Method of Characteristicscited by 1
  7. 2026Koopman operator theory: fundamentals, control, and applicationscited by 1
  8. 2026Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operatorscited by 1
  9. 2023Featurizing Koopman Mode Decomposition For Robust Forecastingcited by 0
  10. 2023Simplicity bias, algorithmic probability, and the random logistic mapcited by 0
  11. 2024Deep Koopman-layered Model with Universal Property Based on Toeplitz Matricescited by 0
  12. 2025Nonparametric Sparse Online Learning of the Koopman Operatorcited by 0
  13. 2025Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operatorscited by 0
  14. 2025Bayesian Transfer Operators in Reproducing Kernel Hilbert Spacescited by 0
  15. 2025Conformal Online Learning of Deep Koopman Linear Embeddingscited by 0
  16. 2025A Dynamics-Informed Gaussian Process Framework for 2D Stochastic Navier-Stokes via Quasi-Gaussianitycited by 0
  17. 2026Adaptive Kernel Selection for Kernelized Diffusion Mapscited by 0
  18. 2026Dictionary learning for Kernel EDMDcited by 0
  19. 2026Learning to control switching nonlinear systems with Koopman operator regressioncited by 0