In recent years, more and more works have appeared devoted to the analog (hardware) implementation of artificial neural networks, in which neurons and the connection between them are based not on computer calculations, but on physical principles. Such networks offer improved energy efficiency and, in some cases, scalability, but may be susceptible to internal noise. This paper studies the influence of noise on the functioning of recurrent networks using the example of trained echo state networks (ESNs). The most common reservoir connection matrices were chosen as various topologies of ESNs: random uniform and band matrices with different connectivity. White Gaussian noise was chosen as the influence, and according to the way of its introducing it was additive or multiplicative, as well as correlated or uncorrelated. In the paper, we show that the propagation of noise in reservoir is mainly controlled by the statistical properties of the output connection matrix, namely the mean and the mean square. Depending on these values, more correlated or uncorrelated noise accumulates in the network. We also show that there are conditions under which even noise with an intensity of $10^{-20}$ is already enough to completely lose the useful signal. In the article we show which types of noise are most critical for networks with different activation functions (hyperbolic tangent, sigmoid and linear) and if the network is self-closed.

翻译：近年来，越来越多的工作致力于人工神经网络的模拟（硬件）实现，其中神经元及其连接并非基于计算机计算，而是基于物理原理。此类网络可提升能效，某些情况下还具有可扩展性，但可能易受内部噪声影响。本文以训练后的回声状态网络（ESNs）为例，研究噪声对循环网络功能的影响。我们选择了ESNs中常见的储层连接矩阵作为不同拓扑结构：随机均匀矩阵和不同连接度的带状矩阵。噪声影响选定为白高斯噪声，根据引入方式分为加性或乘性，以及相关或不相关。在文中，我们证明储层中的噪声传播主要受输出连接矩阵的统计特性控制，即均值和均方值。根据这些值，网络会积累更多相关或不相关噪声。我们还表明，存在某些条件，即使强度为$10^{-20}$的噪声也足以完全丢失有用信号。文章进一步展示了对于具有不同激活函数（双曲正切、sigmoid和线性）的网络，以及当网络自闭合时，哪些类型的噪声最为关键。