Uncovering differential equations from data with hidden variables
This work addresses a domain-specific problem for researchers and practitioners in dynamical systems modeling, offering an incremental improvement over existing SINDy methods.
The paper tackles the problem of learning differential equations from data when some variables are unobserved, extending the SINDy method to handle hidden variables by regressing higher-order derivatives onto a dictionary of lower-order ones. The result is a method that provides high-quality short-term forecasts, is orders of magnitude faster than competing approaches, and is validated on synthetic data and real temperature time series.
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 Réseau de Transport d'Électricité (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.