Reservoir computing, using nonlinear dynamical systems, offers a cost-effective alternative to neural networks for complex tasks involving processing of sequential data, time series modeling, and system identification. Echo state networks (ESNs), a type of reservoir computer, mirror neural networks but simplify training. They apply fixed, random linear transformations to the internal state, followed by nonlinear changes. This process, guided by input signals and linear regression, adapts the system to match target characteristics, reducing computational demands. A potential drawback of ESNs is that the fixed reservoir may not offer the complexity needed for specific problems. While directly altering (training) the internal ESN would reintroduce the computational burden, an indirect modification can be achieved by redirecting some output as input. This feedback can influence the internal reservoir state, yielding ESNs with enhanced complexity suitable for broader challenges. In this paper, we demonstrate that by feeding some component of the reservoir state back into the network through the input, we can drastically improve upon the performance of a given ESN. We rigorously prove that, for any given ESN, feedback will almost always improve the accuracy of the output. For a set of three tasks, each representing different problem classes, we find that with feedback the average error measures are reduced by $30\%-60\%$. Remarkably, feedback provides at least an equivalent performance boost to doubling the initial number of computational nodes, a computationally expensive and technologically challenging alternative. These results demonstrate the broad applicability and substantial usefulness of this feedback scheme.

翻译：储层计算利用非线性动力系统，为涉及序列数据处理、时间序列建模和系统辨识的复杂任务提供了一种成本效益优于神经网络的替代方案。回声状态网络作为储层计算的一种实现形式，在保留神经网络架构的同时显著简化了训练过程。该网络对内部状态施加固定的随机线性变换，继而进行非线性转换。这一过程在输入信号和线性回归的引导下，使系统能够适配目标特性，从而降低计算需求。ESN的潜在缺陷在于固定储层结构可能无法为特定问题提供足够的复杂度。虽然直接调整（训练）ESN内部参数将重新引入计算负担，但通过将部分输出重定向为输入可实现间接修正。这种反馈机制能够影响内部储层状态，从而生成具备增强复杂度、适用于更广泛挑战的ESN。本文论证了通过将储层状态的某些分量反馈至网络输入端，能够显著提升给定ESN的性能。我们严格证明了对于任意给定的ESN，反馈机制几乎总能提高输出精度。在代表不同问题类别的三项任务中，反馈使平均误差指标降低了$30\%-60\%$。值得注意的是，反馈带来的性能提升至少等同于将初始计算节点数量翻倍的效果，而后者是计算成本高昂且技术实现困难的替代方案。这些结果充分证明了该反馈方案具有广泛的适用性和显著的实用价值。