We introduce a data-driven model approximation method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method is based on embedding the nonlinear system in a high (or infinite) dimensional reproducing kernel Hilbert space (RKHS) where linear balanced truncation may be carried out implicitly. This leads to a nonlinear reduction map which can be combined with a representation of the system belonging to a RKHS to give a closed, reduced order dynamical system which captures the essential input-output characteristics of the original model. Working in RKHS provides a convenient, general functional-analytical framework for theoretical understanding. Empirical simulations illustrating the approach are also provided. 1 2 U Z −1 . We also note that the problem of finding the coordinate change Q can be seen as an optimization problem [1] of the form min Q trace[QW c Q * + Q − * W o Q −1 ]. 5

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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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