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

arXiv: https://arxiv.org/abs/2005.13231 · HTML: https://mlsanity.com/paper/2005.13231/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2020.

## Built on by (most cited first, 30 of 31)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Learning to Forecast Dynamical Systems from Streaming Data](https://mlsanity.com/paper/2109.09703/) (2021)
- [Learning invariant representations of time-homogeneous stochastic dynamical systems](https://mlsanity.com/paper/2307.09912/) (2023)
- [One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels](https://mlsanity.com/paper/2209.12086/) (2022)
- [Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems](https://mlsanity.com/paper/2301.10321/) (2023)
- [Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series](https://mlsanity.com/paper/2111.13037/) (2021)
- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [On the Universal Transformation of Data-Driven Models to Control Systems](https://mlsanity.com/paper/2102.04722/) (2021)
- [Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry](https://mlsanity.com/paper/2103.17233/) (2021)
- [Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator](https://mlsanity.com/paper/2412.16588/) (2024)
- [Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approach](https://mlsanity.com/paper/2408.06465/) (2024)
- [Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds](https://mlsanity.com/paper/2603.15091/) (2026)
- [Bridging Algorithmic Information Theory and Machine Learning: A New Approach to Kernel Learning](https://mlsanity.com/paper/2311.12624/) (2023)
- [Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om method](https://mlsanity.com/paper/2403.02811/) (2024)
- [Nonparametric Sparse Online Learning of the Koopman Operator](https://mlsanity.com/paper/2405.07432/) (2024)
- [Ensemble forecasts in reproducing kernel Hilbert space family](https://mlsanity.com/paper/2207.14653/) (2022)
- [Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation](https://mlsanity.com/paper/2603.01077/) (2026)
- [Kernel 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 Characteristics](https://mlsanity.com/paper/2603.06872/) (2026)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators](https://mlsanity.com/paper/2608.06155/) (2026)
- [Featurizing Koopman Mode Decomposition For Robust Forecasting](https://mlsanity.com/paper/2312.09146/) (2023)
- [Simplicity bias, algorithmic probability, and the random logistic map](https://mlsanity.com/paper/2401.00593/) (2023)
- [Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices](https://mlsanity.com/paper/2410.02199/) (2024)
- [Nonparametric Sparse Online Learning of the Koopman Operator](https://mlsanity.com/paper/2501.16489/) (2025)
- [Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operators](https://mlsanity.com/paper/2505.06806/) (2025)
- [Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2509.22482/) (2025)
- [Conformal Online Learning of Deep Koopman Linear Embeddings](https://mlsanity.com/paper/2511.12760/) (2025)
- [A Dynamics-Informed Gaussian Process Framework for 2D Stochastic Navier-Stokes via Quasi-Gaussianity](https://mlsanity.com/paper/2511.21281/) (2025)
- [Adaptive Kernel Selection for Kernelized Diffusion Maps](https://mlsanity.com/paper/2604.18402/) (2026)
- [Dictionary learning for Kernel EDMD](https://mlsanity.com/paper/2604.25572/) (2026)

## 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.