  We develop a deep convolutional neural network (DCNN) based framework for
model-free prediction of the occurrence of extreme events both in time ("when")
and in space ("where") in nonlinear physical systems of spatial dimension two.
The measurements or data are a set of two-dimensional snapshots or images. For
a desired time horizon of prediction, a proper labeling scheme can be
designated to enable successful training of the DCNN and subsequent prediction
of extreme events in time. Given that an extreme event has been predicted to
occur within the time horizon, a space-based labeling scheme can be applied to
predict, within certain resolution, the location at which the event will occur.
We use synthetic data from the 2D complex Ginzburg-Landau equation and
empirical wind speed data of the North Atlantic ocean to demonstrate and
validate our machine-learning based prediction framework. The trade-offs among
the prediction horizon, spatial resolution, and accuracy are illustrated, and
the detrimental effect of spatially biased occurrence of extreme event on
prediction accuracy is discussed. The deep learning framework is viable for
predicting extreme events in the real world.
