  This article presents a general framework for recovering missing dynamical
systems using available data and machine learning techniques. The proposed
framework reformulates the prediction problem as a supervised learning problem
to approximate a map that takes the memories of the resolved and identifiable
unresolved variables to the missing components in the resolved dynamics. We
demonstrate the effectiveness of the proposed framework with a theoretical
guarantee of a path-wise convergence of the resolved variables up to finite
time and numerical tests on prototypical models in various scientific domains.
These include the 57-mode barotropic stress models with multiscale interactions
that mimic the blocked and unblocked patterns observed in the atmosphere, the
nonlinear Schr\"{o}dinger equation which found many applications in physics
such as optics and Bose-Einstein-Condense, the Kuramoto-Sivashinsky equation
which spatiotemporal chaotic pattern formation models trapped ion mode in
plasma and phase dynamics in reaction-diffusion systems. While many machine
learning techniques can be used to validate the proposed framework, we found
that recurrent neural networks outperform kernel regression methods in terms of
recovering the trajectory of the resolved components and the equilibrium
one-point and two-point statistics. This superb performance suggests that
recurrent neural networks are an effective tool for recovering the missing
dynamics that involves approximation of high-dimensional functions.
