标题： 基于状态空间系统的随机动力学学习 显示英文标题

标题： Stochastic dynamics learning with state-space systems

摘要： 这项工作通过在确定性和随机设置中对褪色记忆和回声状态属性（ESP）提供统一的处理，推进了储备计算（RC）的理论基础。我们研究了状态空间系统，这是时间序列学习中的核心模型类，并确立了褪色记忆和解稳定性通常是成立的——即使在没有ESP的情况下也是如此——为RC模型的实证成功提供了稳健的解释，而无需严格的压缩条件。在随机情况下，我们对随机回声状态进行了关键评估，提出了一种基于概率分布空间上的吸引子动力学的新型分布视角，这导致了一个丰富且连贯的理论。我们的结果扩展并推广了之前关于非自治动力系统的工作，为RC模型中的因果关系、稳定性和记忆提供了新的见解。这为在确定性和随机 regimes 中可靠的时间数据生成建模奠定了基础。 显示英文摘要

摘要： This work advances the theoretical foundations of reservoir computing (RC) by providing a unified treatment of fading memory and the echo state property (ESP) in both deterministic and stochastic settings. We investigate state-space systems, a central model class in time series learning, and establish that fading memory and solution stability hold generically -- even in the absence of the ESP -- offering a robust explanation for the empirical success of RC models without strict contractivity conditions. In the stochastic case, we critically assess stochastic echo states, proposing a novel distributional perspective rooted in attractor dynamics on the space of probability distributions, which leads to a rich and coherent theory. Our results extend and generalize previous work on non-autonomous dynamical systems, offering new insights into causality, stability, and memory in RC models. This lays the groundwork for reliable generative modeling of temporal data in both deterministic and stochastic regimes.