Uncovering differential equations from data with hidden variables

TL;DR

This paper extends the SINDy method to learn differential equations with hidden variables, enabling accurate short-term predictions and significantly faster computation on synthetic and real temperature data.

Contribution

The authors introduce a novel extension of SINDy that handles unobserved variables by regressing higher order derivatives, improving speed and prediction accuracy.

Findings

High-quality short-term forecasts achieved

Orders of magnitude faster than competing methods

Effective on both synthetic and real temperature data

Abstract

SINDy is a method for learning system of differential equations from data by solving a sparse linear regression optimization problem [Brunton et al., 2016]. In this article, we propose an extension of the SINDy method that learns systems of differential equations in cases where some of the variables are not observed. Our extension is based on regressing a higher order time derivative of a target variable onto a dictionary of functions that includes lower order time derivatives of the target variable. We evaluate our method by measuring the prediction accuracy of the learned dynamical systems on synthetic data and on a real data-set of temperature time series provided by the R\'eseau de Transport d'\'Electricit\'e (RTE). Our method provides high quality short-term forecasts and it is orders of magnitude faster than competing methods for learning differential equations with latent variables.