Uncovering differential equations from data with hidden variables

SINDy is a method for learning system of differential equations from data by solving a sparse linear regression optimization problem [Brunton, Proctor, and Kutz, Proc. Natl. Acad. Sci. USA 113, 3932 (2016)PNASA60027-842410.1073/pnas.1517384113]. 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 Réseau de Transport d'Électricité. 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.