  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 [Floryan, D. & Graham, M. D.
Data-driven discovery of intrinsic dynamics. Nat Mach Intell $\textbf{4}$,
1113-1120 (2022)], here denoted as $\textit{Charts and Atlases for Nonlinear
Data-Driven Dynamics on Manifolds}$, 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 [Moehlis, J., Faisst, H. & Eckhardt, B. A low-dimensional
model for turbulent shear flows. New J Phys $\textbf{6}$, 56 (2004)], which
undergoes extreme events in the form of intermittent quasi-laminarization and
long-time full laminarization. We demonstrate that the CANDyMan method allows
the trained dynamical models to more accurately forecast the evolution of the
model coefficients, reducing the error in the predictions as the model evolves
forward in time. The technique exhibits more accurate predictions of extreme
events, capturing the frequency of quasi-laminarization events and predicting
the time until full laminarization more accurately than a single neural
network.
