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Dynamical systems with high intrinsic dimensionality are often characterizedby extreme events having the form of rare transitions several standarddeviations away from the mean. For such systems, order-reduction methodsthrough projection of the governing equations have limited applicability due tothe large intrinsic dimensionality of the underlying attractor but also thecomplexity of the transient events. An alternative approach is data-driventechniques that aim to quantify the dynamics of specific modes utilizingdata-streams. Several of these approaches have improved performance byexpanding the state representation using delayed coordinates. However, suchstrategies are limited in regions of the phase space where there is a smallamount of data available, as is the case for extreme events. In this work, wedevelop a blended framework that integrates an imperfect model, obtained fromprojecting equations into a subspace that still contains crucial dynamicalinformation, with data-streams through a recurrent neural network (RNN)architecture. In particular, we employ the long-short-term memory (LSTM), tomodel portions of the dynamics which cannot be accounted by the equations. TheRNN is trained by analyzing the mismatch between the imperfect model and thedata-streams, projected in the reduced-order space. In this way, thedata-driven model improves the imperfect model in regions where data isavailable, while for locations where data is sparse the imperfect model stillprovides a baseline for the prediction of the system dynamics. We assess thedeveloped framework on two challenging prototype systems exhibiting extremeevents and show that the blended approach has improved performance comparedwith methods that use either data streams or the imperfect model alone. Theimprovement is more significant in regions associated with extreme events,where data is sparse.

Data-assisted reduced-order modeling of extreme events in complex dynamical systems