破解递归神经网络记忆机制，统一解释确定与随机场景下的稳定性能。
Stochastic dynamics learning with state-space systems
- 基于状态空间系统建模，无需严格收缩条件即可保证记忆衰减与解的稳定性。
- 在随机情形下提出分布吸引子新视角，构建更完整的可解释理论框架。
- 适合研究时序建模、动态系统理论及生成模型的学者深入参考。
本文从理论层面推进了水库计算（RC）的发展，对确定与随机设定下的记忆衰减和回声状态性质（ESP）提供了统一处理。研究聚焦时间序列学习的核心模型——状态空间系统，证明记忆衰减与解的稳定性在普遍条件下成立，即使不满足ESP也依然有效，为RC模型的实证成功提供了稳健解释。在随机情形下，我们批判性评估了随机回声状态，并提出一种基于概率分布空间吸引子动力学的新分布视角，构建出丰富且自洽的理论体系。结果拓展并推广了此前关于非自治动力系统的成果，深化了对因果性、稳定性与记忆机制的理解。本工作为确定与随机两种情形下时序数据的可靠生成建模奠定了基础。
原文摘要 · Abstract (English)
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.
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