Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series

TL;DR

This paper introduces a method to improve data-driven learning of dynamical systems from irregularly-sampled time series by incorporating time differences into kernel-based vector field interpolation, enhancing forecasting accuracy.

Contribution

It proposes a novel approach that directly accounts for irregular sampling in kernel-based vector field learning, improving performance over classical methods.

Findings

Significant improvement in forecasting accuracy on benchmark systems

Method remains simple, fast, and robust

Effective for irregularly-sampled time series data

Abstract

A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a kernel. In particular, this strategy is highly efficient (both in terms of accuracy and complexity) when the kernel is data-adapted using Kernel Flows (KF)\cite{Owhadi19} (which uses gradient-based optimization to learn a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used for interpolation). Despite its previous successes, this strategy (based on interpolating the vector field driving the dynamical system) breaks down when the observed time series is not regularly sampled in time. In this work, we propose to address this problem by directly approximating the vector field of the dynamical system by incorporating time differences between observations in the (KF) data-adapted kernels. We compare our approach with the classical one over different benchmark dynamical systems and show that it significantly improves the forecasting accuracy while remaining simple, fast, and robust.