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.

翻译：回声状态网络（ESN）是一种循环神经网络，在表示时间序列和非线性动态系统方面展现出良好前景。尽管其具备高效的训练流程，但这类储层计算策略（如ESN）需要高阶网络（即大量神经元），导致其状态数量远高于模型输入与输出维度。大量状态不仅增加时间步计算成本，还可能引发鲁棒性问题，尤其在将ESN应用于模型预测控制（MPC）及其他最优控制问题时。规避该复杂性问题的途径之一是采用模型降阶策略，如本征正交分解（POD）及其变体（POD-DEIM），通过为已训练的高维ESN寻找等效低维表示。为此，本文旨在研究并分析POD方法在回声状态网络中的性能，通过比较POD降阶网络与原始（全阶）ESN的记忆容量（MC）评估其有效性。我们还对两个数值案例进行了实验：一个NARMA10差分方程系统及一个包含两口油井和一根立管的石油平台。结果表明，原始ESN与POD降阶网络相比性能损失极小，且POD降阶ESN的性能通常优于相同尺寸的常规ESN。此外，与原始ESN相比，POD降阶网络实现了约80%的运算加速。