Predicting extreme events in a data-driven model of turbulent shear flow using an atlas of charts

Dynamical systems with extreme events are difficult to capture with data-driven modeling due to the relative scarcity of data within extreme events compared to the typical dynamics of the system and the strong dependence of the long-time occurrence of extreme events on short-time conditions. A recently developed technique [D. Floryan and M. D. Graham, Nat. Mach. Intell. 4, 1113 (2022)2522-583910.1038/s42256-022-00575-4], here denoted as Charts and Atlases for Nonlinear Data-Driven Dynamics on Manifolds, or CANDyMan, overcomes these difficulties 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 to yield 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 [J. Moehlis, H. Faisst, and B. Eckhardt, New J. Phys. 6, 56 (2004)1367-263010.1088/1367-2630/6/1/056], which undergoes extreme events in the form of intermittent quasi-laminarization and long-time full laminarization. The multichart model created by the CANDyMan technique is compared with both a standard data-driven model (i.e., the "single-chart"limit of the CANDyMan method) and a Koopman-based model created through extended dynamic mode decomposition-dictionary learning. We demonstrate that the CANDyMan method allows the trained dynamical models to more accurately forecast the evolution of the model coefficients than both a single-chart model and a Koopman model, reducing the error in the predictions as the model evolves forward in time. The technique exhibits more accurate predictions of extreme events than either a single-chart model or Koopman model, capturing the frequency of quasi-laminarization events and predicting the time until full laminarization more accurately than a single neural network.