Effective models and predictability of chaotic multiscale systems via machine learning.

Understanding and modeling the dynamics of multiscale systems is a problem of considerable interest both for theory and applications. For unavoidable practical reasons, in multiscale systems, there is the need to eliminate from the description the fast and small-scale degrees of freedom and thus build effective models for only the slow and large-scale degrees of freedom. When there is a wide scale separation between the degrees of freedom, asymptotic techniques, such as the adiabatic approximation, can be used for devising such effective models, while away from this limit there exist no systematic techniques. Here, we scrutinize the use of machine learning, based on reservoir computing, to build data-driven effective models of multiscale chaotic systems. We show that, for a wide scale separation, machine learning generates effective models akin to those obtained using multiscale asymptotic techniques and, remarkably, remains effective in predictability also when the scale separation is reduced. We also show that predictability can be improved by hybridizing the reservoir with an imperfect model.