Kernel-based approximation of the Koopman generator and Schrödinger operator
This work provides a method for dimensionality reduction in dynamical systems and quantum mechanics, enabling cross-application of techniques between stochastic differential equations and quantum systems, but it appears incremental as it builds on existing kernel and operator approximation frameworks.
The authors tackled the problem of approximating differential operators like the Koopman generator and Schrödinger operator from data, proposing a kernel-based method that estimates eigenfunctions via auxiliary matrix eigenvalue problems, with applications in molecular dynamics and quantum chemistry.
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.