  A machine-learning approach called "reservoir computing" has been used
successfully for short-term prediction and attractor reconstruction of chaotic
dynamical systems from time series data. We present a theoretical framework
that describes conditions under which reservoir computing can create an
empirical model capable of skillful short-term forecasts and accurate long-term
ergodic behavior. We illustrate this theory through numerical experiments. We
also argue that the theory applies to certain other machine learning methods
for time series prediction.
