  Dynamical systems with high intrinsic dimensionality are often characterized
by extreme events having the form of rare transitions several standard
deviations away from the mean. For such systems, order-reduction methods
through projection of the governing equations have limited applicability due to
the large intrinsic dimensionality of the underlying attractor but also the
complexity of the transient events. An alternative approach is data-driven
techniques that aim to quantify the dynamics of specific modes utilizing
data-streams. Several of these approaches have improved performance by
expanding the state representation using delayed coordinates. However, such
strategies are limited in regions of the phase space where there is a small
amount of data available, as is the case for extreme events. In this work, we
develop a blended framework that integrates an imperfect model, obtained from
projecting equations into a subspace that still contains crucial dynamical
information, with data-streams through a recurrent neural network (RNN)
architecture. In particular, we employ the long-short-term memory (LSTM), to
model portions of the dynamics which cannot be accounted by the equations. The
RNN is trained by analyzing the mismatch between the imperfect model and the
data-streams, projected in the reduced-order space. In this way, the
data-driven model improves the imperfect model in regions where data is
available, while for locations where data is sparse the imperfect model still
provides a baseline for the prediction of the system dynamics. We assess the
developed framework on two challenging prototype systems exhibiting extreme
events and show that the blended approach has improved performance compared
with methods that use either data streams or the imperfect model alone. The
improvement is more significant in regions associated with extreme events,
where data is sparse.
