DeepAI AI Chat
Log In Sign Up

Kernel-based approximation of the Koopman generator and Schrödinger operator

by   Stefan Klus, et al.

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.


page 1

page 2

page 3

page 4


Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces

Transfer operators such as the Perron-Frobenius or Koopman operator play...

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

We derive a data-driven method for the approximation of the Koopman gene...

The sine kernel, two corresponding operator identities, and random matrices

In the present paper, we consider the integral operator, which acts in H...

An Introduction to Kernel and Operator Learning Methods for Homogenization by Self-consistent Clustering Analysis

Recent advances in operator learning theory have improved our knowledge ...

Efficient Quantum Computation of the Fermionic Boundary Operator

The boundary operator is a linear operator that acts on a collection of ...

Hierarchical regularization networks for sparsification based learning on noisy datasets

We propose a hierarchical learning strategy aimed at generating sparse r...