Abstract
The prediction of extreme events, from avalanches and droughts to tsunamis and epidemics, depends on the formulation and analysis of relevant, complex dynamical systems. Such dynamical systems are characterized by high intrinsic dimensionality with extreme events having the form of rare transitions that are several standard deviations away from the mean. Such systems are not amenable to classical order-reduction methods through projection of the governing equations due to the large intrinsic dimensionality of the underlying attractor as well as the complexity of the transient events. Alternatively, data-driven techniques aim to quantify the dynamics of specific, critical modes by utilizing data-streams and by expanding the dimensionality of the reduced-order model using delayed coordinates. In turn, these methods have major limitations in regions of the phase space with sparse data, which is the case for extreme events. In this work, we develop a novel hybrid framework that complements an imperfect reduced order model, with data-streams that are integrated though a recurrent neural network (RNN) architecture. The reduced order model has the form of projected equations into a low-dimensional subspace that still contains important dynamical information about the system and it is expanded by a long short-term memory (LSTM) regularization. The LSTM-RNN is trained by analyzing the mismatch between the imperfect model and the data-streams, projected to the reduced-order space. The data-driven model assists 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 state. We assess the developed framework on two challenging prototype systems exhibiting extreme events. We show that the blended approach has improved performance compared with methods that use either data streams or the imperfect model alone. Notably the improvement is more significant in regions associated with extreme events, where data is sparse.
Highlights
Extreme events are omnipresent in important problems in science and technology such as turbulent and reactive flows [1, 2], Kolmogorov [3] and unstable plane Couette flow [4]), geophysical systems, nonlinear optics [9, 10] or water waves [11,12,13]), and mechanical systems.The complete description of these system through the governing equations is often challenging either because it is very hard/expensive to solve these equations with an appropriate resolution or due to the magnitude of the model errors
We focus on data-driven recurrent neural networks (RNN) with a long-short term memory (LSTM) [30] that represents some of the truncated degrees-of-freedom
We introduce a data assisted framework for reduced-order modeling and prediction of extreme transient events in complex dynamical systems with high-dimensional attractors
Summary
Extreme events are omnipresent in important problems in science and technology such as turbulent and reactive flows [1, 2], Kolmogorov [3] and unstable plane Couette flow [4]), geophysical systems (e.g. climate dynamics [5, 6], cloud formations in tropical atmospheric convection [7, 8]), nonlinear optics [9, 10] or water waves [11,12,13]), and mechanical systems (e.g. mechanical metamaterials [14, 15]).The complete description of these system through the governing equations is often challenging either because it is very hard/expensive to solve these equations with an appropriate resolution or due to the magnitude of the model errors. The very large dimensionality of their attractor in combination with the occurrence of important transient, but rare events, makes the application of classical order-reduction methods a challenging task. The same limitations hold for data-driven, parametric methods [26,27,28,29], where the assumed analytical representations have parameters that are optimized so that the resulted model best fits the data. These methods perform well when the system operates within the main ‘core’ of the attractor, this may not be the case when rare and/or extreme events occur
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