  To predict rare extreme events using deep neural networks, one encounters the
so-called small data problem because even long-term observations often contain
few extreme events. Here, we investigate a model-assisted framework where the
training data is obtained from numerical simulations, as opposed to
observations, with adequate samples from extreme events. However, to ensure the
trained networks are applicable in practice, the training is not performed on
the full simulation data; instead we only use a small subset of observable
quantities which can be measured in practice. We investigate the feasibility of
this model-assisted framework on three different dynamical systems (Rossler
attractor, FitzHugh-Nagumo model, and a turbulent fluid flow) and three
different deep neural network architectures (feedforward, long short-term
memory, and reservoir computing). In each case, we study the prediction
accuracy, robustness to noise, reproducibility under repeated training, and
sensitivity to the type of input data. In particular, we find long short-term
memory networks to be most robust to noise and to yield relatively accurate
predictions, while requiring minimal fine-tuning of the hyperparameters.
