Kernel-based approximation of the Koopman generator and Schrödinger operator

Abstract

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 Schrödinger 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 Schrödinger 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.

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Cited by in corpus (16)

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• Learning dynamical systems from data: a simple cross-validation perspective
• Koopman-based feedback design with stability guarantees
• Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem
• One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels
• An overview of Koopman-based control: From error bounds to closed-loop guarantees
• Efficient Approximation of Molecular Kinetics using Random Fourier Features
• Koopman analysis of quantum systems
• Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry
• Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series
• Kernel-based error bounds of bilinear Koopman surrogate models for nonlinear data-driven control
• tgEDMD: Approximation of the Kolmogorov Operator in Tensor Train Format
• Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator
• On the Universal Transformation of Data-Driven Models to Control Systems
• Dynamical systems and complex networks: A Koopman operator perspective
• Linear quadratic control of nonlinear systems with Koopman operator learning and the Nyström method