Uncovering differential equations from data with hidden variables

Examples include meteorology, biology, and physics. The usual way to model deterministic dynamical systems is by using (partial) differential equations. Typically, differential equations models for a given dynamical system are derived using apriori insights into the problem at hand; then the model is validated using empirical observations. In an era in which massive data-sets pertaining to different fields of science are widely available, an interesting problem is whether it is possible for a useful differential equations model to be learned directly from data, without any major modeling effort required by the researcher. Our goal in this paper is to develop a general methodology for building such differential equations models in contexts in which not all relevant variables are observed, that is, in cases in which the main variable of interest depends on other variables of which no measurements are available. As a concrete example, consider the following problem. RTE, the electricity transmission system operator of France, uses high-level simulations of hourly temperature series to study the impact different climate scenarios have on electricity consumption, and hence on the French electrical power grid.