Data-driven modeling of a dynamic system with extreme events through neural networks in an atlas of charts

Fluid dynamic systems with extreme events are difficult to capture with data-driven modeling, due to strong dependence of the long-time occurrence of extreme events on short-time conditions and the relative scarcity of data within extreme events compared to non-extreme states. Our technique known as Charts and Atlases for Nonlinear Data-Driven Dynamics on Manifolds, or CANDyMan, works by decomposing the time series into separate charts based on data similarity, learning dynamical models on each chart via individual time-mapping neural networks, then stitching the charts together to create a single atlas, obtaining a global dynamical model. We apply CANDyMan to a nine-dimensional model of turbulent shear flow between infinite parallel free-slip walls under a sinusoidal body force developed by Moehlis, Faisst and Eckhart (MFE), which undergoes extreme events in the form of high-energy, intermittent quasi-relaminarization. We demonstrate that the application of CANDyMan reduces the error in predictions and captures the frequency of high-energy extreme events more accurately than a single time-mapping neural network. Finally, we project onto the full velocity field, where CANDyMan creates a more accurate reproduction of the turbulent velocity statistics than a single dynamical model.