A data-driven framework for non-stationary complex systems: Blending generalized Langevin and neural ordinary-differential equations.*

Complex systems (CSs) are ubiquitous in nature and technology. Understanding the nature of CSs requires of a precise description of the time evolution of their observables. Existing mathematical models pose considerable challenges, e.g. requiring a priori understanding of the prevailing dynamics of the CS at hand which, in turn, limits their applicability, especially when the CS alternates between stationary and non-stationary behavior. We propose here a general framework for CSs which addresses these challenges. Its cornerstone is a data-driven model which combines a generalized Langevin equation (GLE) and a neural ordinary-differential equation (ODE). The GLE captures fine-grained details of the CS behavior under the stationarity assumption, while the neural ODE reconstructs and forecasts the differences between the actual dynamics and the trajectories generated by the GLE, thus approximating the non-stationary features not accounted for by the GLE. Our framework enables the rational and systematic scrutiny of CSs exhibiting both stationary and non-stationary features, and we exemplify its effectiveness and robustness using empirical data from the energy sector, specifically the Spanish electricity day-ahead market.