Learning dynamical systems from data: A simple cross-validation perspective, part II: Nonparametric kernel flows

Abstract

In previous work, we showed that learning dynamical system Hamzi and Owhadi (2021) with kernel methods can achieve state of the art, both in terms of accuracy and complexity, for predicting climate/weather time series Hamzi et al., (2021) as well as for a family of 133 chaotic systems Lu et al., (2023), Yang et al., (2024), when the kernel is also learned from data. While the kernels considered in previous work were parametric, in this follow-up paper, we test a non-parametric approach and tune warping kernels (with kernel flows, a variant of cross-validation) for learning prototypical dynamical systems. We train the kernel using the regression relative error between two interpolants (measured in the RKHS norm of the kernel) as the quantity to minimize, as well as using the Maximum Mean Discrepancy between two different samples, and that characterizes the statistical properties of the dynamical system, as a the quantity to minimize.