  Reservoir computing (RC), a particular form of recurrent neural network, is
under explosive development due to its exceptional efficacy and high
performance in reconstruction or/and prediction of complex physical systems.
However, the mechanism triggering such effective applications of RC is still
unclear, awaiting deep and systematic exploration. Here, combining the delayed
embedding theory with the generalized embedding theory, we rigorously prove
that RC is essentially a high dimensional embedding of the original input
nonlinear dynamical system. Thus, using this embedding property, we unify into
a universal framework the standard RC and the time-delayed RC where we novelly
introduce time delays only into the network's output layer, and we further find
a trade-off relation between the time delays and the number of neurons in RC.
Based on this finding, we significantly reduce the network size of RC for
reconstructing and predicting some representative physical systems, and, more
surprisingly, only using a single neuron reservoir with time delays is
sometimes sufficient for achieving those tasks.
