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：利用非线性动力系统的储层计算，为处理序列数据、时间序列建模和系统辨识等复杂任务提供了一种经济高效的神经网络替代方案。回声状态网络（ESNs）作为储层计算机的一种，虽模拟神经网络但简化了训练过程。其对内部状态施加固定的随机线性变换，并进行非线性变换。该过程在输入信号和线性回归的引导下，使系统适应目标特征，从而降低计算需求。ESNs的潜在缺陷在于固定储层可能无法提供特定问题所需的复杂度。虽然直接改变（训练）内部ESN会重新引入计算负担，但通过将部分输出重新定向为输入可实现间接修改。这种反馈能影响内部储层状态，生成具有增强复杂度的ESNs以应对更广泛挑战。本文证明，通过将储层状态的某些分量经输入反馈回网络，可显著提升给定ESN的性能。我们严格论证：对任意ESN而言，反馈几乎总能提高输出精度。在分别代表不同问题类别的三项任务中，我们发现采用反馈后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相当于将初始计算节点数量翻倍——而这正是计算代价高昂且技术实现困难的替代方案。这些结果证明了该反馈方案的广泛适用性与显著实用价值。