Uncovering differential equations from data with hidden variables

Finding a set of differential equations to model dynamical systems is a difficult task present in many branches of science and engineering. We propose a method to learn systems of differential equations directly from data. Our method is based on solving a tailor-made $\ell_1-$regularised least-squares problem and can deal with latent variables by adding higher-order derivatives to account for the lack of information. Extensive numerical studies show that our method can recover useful representations of the dynamical system that generated the data even when some variables are not observed. Moreover, being based on solving a convex optimisation problem, our method is much faster than competing approaches based on solving combinatorial problems. Finally, we apply our methodology to a real data-set of temperature time series.