  Echo State Networks (ESN) are a type of Recurrent Neural Network that yields
promising results in representing time series and nonlinear dynamic systems.
Although they are equipped with a very efficient training procedure, Reservoir
Computing strategies, such as the ESN, require high-order networks, i.e., many
neurons, resulting in a large number of states that are magnitudes higher than
the number of model inputs and outputs. A large number of states not only makes
the time-step computation more costly but also may pose robustness issues,
especially when applying ESNs to problems such as Model Predictive Control
(MPC) and other optimal control problems. One way to circumvent this complexity
issue is through Model Order Reduction strategies such as the Proper Orthogonal
Decomposition (POD) and its variants (POD-DEIM), whereby we find an equivalent
lower order representation to an already trained high dimension ESN. To this
end, this work aims to investigate and analyze the performance of POD methods
in Echo State Networks, evaluating their effectiveness through the Memory
Capacity (MC) of the POD-reduced network compared to the original (full-order)
ESN. We also perform experiments on two numerical case studies: a NARMA10
difference equation and an oil platform containing two wells and one riser. The
results show that there is little loss of performance comparing the original
ESN to a POD-reduced counterpart and that the performance of a POD-reduced ESN
tends to be superior to a normal ESN of the same size. Also, the POD-reduced
network achieves speedups of around $80\%$ compared to the original ESN.

Machine Learning;Neural and Evolutionary Computing;Systems and Control;Systems and Control