Reservoir computing is a well-established approach for processing data with a much lower complexity compared to traditional neural networks. Despite two decades of experimental progress, the core properties of reservoir computing (namely separation, robustness, and fading memory) still lack rigorous mathematical foundations. This paper addresses this gap by providing a control-theoretic framework for the analysis of time-delay-based reservoir computers. We introduce formal definitions of the separation property and fading memory in terms of functional norms, and establish their connection to well-known stability notions for time-delay systems as incremental input-to-state stability. For a class of linear reservoirs, we derive an explicit lower bound for the separation distance via Fourier analysis, offering a computable criterion for reservoir design. Numerical results on the NARMA10 benchmark and continuous-time system prediction validate the approach with a minimal digital implementation.

翻译：储层计算是一种相比传统神经网络复杂度更低的成熟数据处理方法。尽管经过二十年的实验进展，储层计算的核心性质（即分离性、鲁棒性和渐逝记忆）仍缺乏严格的数学基础。本文通过为基于时间延迟的储层计算分析提供控制理论框架来填补这一空白。我们以函数范数形式给出了分离性质和渐逝记忆的形式化定义，并建立了它们与时间延迟系统经典稳定性概念（如增量输入到状态稳定性）之间的联系。针对一类线性储层，我们通过傅里叶分析推导出分离距离的显式下界，为储层设计提供了可计算的准则。在NARMA10基准测试和连续时间系统预测上的数值结果验证了该方法的最小化数字实现的有效性。