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

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) [34] (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.

Computational graph completion

“…We have focused the manuscript on the situation where the kernels of the underlying GPs are given/pre-determined. Using data-driven kernels can improve the accuracy of kernel methods by orders of magnitude [34,5,16,15,10,21,35]. There are essentially three categories of methods for learning kernel from data: variants of cross-validation (such as Kernel Flows [34]), maximum likelihood estimation (see [5] for a comparison between Kernel Flows and MLE), and maximum a posteriori estimation [30].…”

Gaussian process hydrodynamics

“…When the underlying physics is unknown, the kernel can be learned from data via crossvalidation/maximum likelihood estimation in a given (possibly non-parametric) family of kernels [25,[37][38] . The kernel flow (a variant of cross-validation) approach [37] has been shown to be efficient for learning (possibly stochastic) dynamical systems [39][40][41][42][43] and designing surrogate models [44][45][46] . In particular, this approach has been shown to compare favorably to ANN-based methods (in terms of both complexity and accuracy) for weather/climate prediction using actual satellite data [40] .…”