KOOPMAN-BASED METHODS AND OPTIMALLY TIME-DEPENDENT MODES FOR THE DATA-DRIVEN ANALYSIS OF EXTREME EVENTS

Abstract

Extreme events (rare and large deviations of an observable from its usual value) on nonlinear dynamical systems are studied from a data-driven perspective. The optimally time-dependent modes can give information about instabilities when the derivative of the update rule is known, justifying the use of local approaches or data-driven differentiable approximation of the update rule to obtain information. On the other side, the Koopman operator, mainly used for prediction may give global information on the system (and such events) but its numerical approximation using the Dynamical Mode Decomposition can fail in this context, and be related to local properties instead (of potential interest for extreme events). We investigate the use of the local Dynamical Mode Decomposition on a dataset, showing the limits of the naive use of locally growing modes as indicators of extreme events and proposing a variant of the algorithm to better account for straightened flows.