Active learning of nonlinear operators via neural nets for forecasting instabilities leading to extreme events in fluids

Through approximations of nonlinear operators via neural networks, we develop a framework for computing predictors of extreme events (i.e. instabilities), in infinite-dimensional systems with applications to fluids. Extreme phenomena or instabilities, such as pandemic spikes, electrical-grid failure, or rogue waves, have catastrophic consequences for society. Unfortunately, characterizing extreme events is difficult because of their rarity of occurrence, the infinite-dimensionality of the dynamics, and the stochastic perturbations that excite them. These challenges are problematic as standard training of machine learning models assumes both plentiful data and moderate dimensionality. Neither is the case for extreme events. To navigate these challenges, we combine a neural network architecture designed for approximating infinite-dimensional, nonlinear operators with novel training schemes that actively select data for characterizing extreme events. We apply and assess these methods to prototype systems for deep-water waves having the form of partial differential equations. In this case, the extreme events take the form of randomly triggered modulation instabilities. Finally, we conclude by discussing the generality of this approach for modeling other extreme phenomena.