  Chimeras and branching are two archetypical complex phenomena that appear in
many physical systems; because of their different intrinsic dynamics, they
delineate opposite non-trivial limits in the complexity of wave motion and
present severe challenges in predicting chaotic and singular behavior in
extended physical systems. We report on the long-term forecasting capability of
Long Short-Term Memory (LSTM) and reservoir computing (RC) recurrent neural
networks, when they are applied to the spatiotemporal evolution of turbulent
chimeras in simulated arrays of coupled superconducting quantum interference
devices (SQUIDs) or lasers, and branching in the electronic flow of
two-dimensional graphene with random potential. We propose a new method in
which we assign one LSTM network to each system node except for "observer"
nodes which provide continual "ground truth" measurements as input; we refer to
this method as "Observer LSTM" (OLSTM). We demonstrate that even a small number
of observers greatly improves the data-driven (model-free) long-term
forecasting capability of the LSTM networks and provide the framework for a
consistent comparison between the RC and LSTM methods. We find that RC requires
smaller training datasets than OLSTMs, but the latter requires fewer observers.
Both methods are benchmarked against Feed-Forward neural networks (FNNs), also
trained to make predictions with observers (OFNNs).
