Learning process and stability analysis of reservoir computing

摘要：

储备池计算已广泛应用于学习和预测动态行为，可以实现包括混沌系统在内的复杂时序信号的演化预测。然而，储备池如同大部分神经网络算法一样，其学习过程被认为是黑盒子，缺乏对其系统性的研究和解释。本文通过建立最小单节点储备池计算模型，将学习与预测过程作为一种非线性动力系统进行分析。研究发现，训练序列与输出序列之间存在复杂的映射关系，且与待学习系统本身的动力学分岔性质密切相关。通过稳定性分析，得到了输入和输出状态之间对应关系的相图，揭示了预测失败现象对应的储备池系统稳定性的变化。在单节点系统分析的基础上，进一步发现了随着储备池规模的增大，预测失败概率的快速降低，以及其对应的储备池稳定参数区域的迅速增大，本文以logistic映射对应的周期与混沌信号为例，理论分析与数值结果完全对应,通过储备池学习过程与稳定性的分析，为算法在学习和预测各种动态行为中的成功提供了动力学视角的理论基础。

Reservoir computing has been widely applied to learning and predicting dynamie behaviors, enabling the pre-diction of complex temporal signals, including chaotic systems. However, like most neural network algorithms, the learning process of reservoir computing is often considered a black box, lacking systematic research and explanation. This paper analyzes the learning and prediction process as a nonlinear dynamical system by establishing a minimal single-node reservoir computing model, The study found that there is a complex mapping relationship between the training sequence and the output sequence,which is closely related to the dynamical bifurcation properties of the system to be learned. Through stability analysis, the phase diagram of the correspondence between input and output states was obtained, revealing the changes in the stability of the reservoir system associated with prediction failure phenomena. Based on the analysis of the single-node system, it was further discovered that as the size of the reservoir increases, the probability of prediction failure decreases rapidly, and the corresponding stability parameter region of the reservoir expands rapidly. Using the logistic map corresponding to periodic and chaotic signals as an example, the theoretical analysis matches the numerical results completely. By analyzing the learning process and stability of the reservoir, this paper provides a theoretical foundation from a dynamical perspective for the success of algorithms in learning and predicting various dynamic behaviors.

作者: 

蓝秀文，陈伟，高健，颜子翔，肖井华

LanXiuwen，Chen Wei，GaoJian，YanZixiang，XiaoJinghua

机构地区：

北京邮电大学理学院;数学与信息网络教育部重点实验室；安徽科技学院信息与网络工程学院

引用本文：

蓝秀文，陈伟，高健等。储备池算法学习过程与稳定性分析[J].河南师范大学学报(自然科学版).2026.54(2):124-134.(Lan Xiuwen, Chen Wei, Gao Jian,et al.Learning process and stability analysis of reservoir computing[J].Journal of Henan Normal University(Natural Science Edition),2026,54(2):124-134.

DOI:10.16366/j.cnki.1000-2367.2025.02.22.0002.)

基金：

国家自然科学基金；安徽省高校科研项目

关键词：

储备池算法；机器学习；非线性动力学；稳定性分析

reservoir computing;machine learning;nonlinear dynamics;stability analysis

分类号：

O415

储备池算法学习过程与稳定性分析.pdf