Hausdorff metric based training of kernels to learn attractors with application to 133 chaotic dynamical systems

Abstract

Kernel methods have been successfully used to learn surrogate models for dynamical systems by regressing the vector field of such systems. However, choosing an appropriate kernel for a specific problem is an essential challenge in practice. Kernel Flows (KFs) have emerged as an effective tool for learning data-adaptive kernels used to interpolate dynamical systems. In this paper, we introduce Hausdorff Metric based Kernel Flows (HMKFs) to discover a’good’ kernel from time series when a system under consideration has an attractor. First, under the premise that a kernel is good if there is no significant loss in accuracy when half of the data is used to reconstruct the attractor, we design an objective function based on the distance between true and forecasted attractors. Then, we optimize the proposal by adding an ℓ1 penalty term when the base kernel contains many terms, thereby generating sparse HMKFs. Furthermore, we apply HMKFs and its sparse version to a library of 133 chaotic systems from various fields such as biochemistry, fluid mechanics, and astrophysics. The results showcase the potential of HMKFs in accurately modeling complex dynamical systems using data-driven kernels.