Learning the Dynamics of Sparsely Observed Interacting Systems

TL;DR

This paper introduces a novel method for learning the dynamics of systems with sparse and irregular observations by framing the problem as a controlled differential equation and using signature theory to enable efficient prediction.

Contribution

It presents a new approach that leverages signature theory and controlled differential equations to learn system dynamics from sparsely observed data, outperforming existing methods.

Findings

Outperforms existing algorithms in recovering full time series.

Provides explicit error bounds dependent on sampling schemes.

Demonstrates effectiveness on real-world epidemiological data.

Abstract

We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.