AI 中文总结
本研究开发离散时间框架，结合SSRC等结构化随机提升方法，通过四个数值实验验证其在概率局域化动力学识别中的有效性，提出满足多要求的最小充分表示。
AI 中文摘要
本研究开发了一种离散时间框架，用于通过适配空间、时间、记忆和状态信息的有限随机表示来识别概率局域化动力学。一个与动力学相关的紧凑集合被有限可测分区局域化，生成可观测概率状态和允许转移的关系图。源自随机结构化储备池计算（SSRC）的结构化随机提升提供了可观测状态的无损多项式表示，而随机延迟提升则添加了有限可观测记忆。这些方法与动力学知情状态空间富集不同：对包含相同当前观测但不同可观测未来的状态的观测纤维进行细化，得到精确的闭合障碍准则。一个路线网络玩具问题给出了最小障碍示例，而四个数值实验室（旋转相位动力学、混沌逻辑映射、范德波尔振荡器和合成循环库存系统）展示了空间尺度、时间尺度、多项式次数和延迟深度的相互作用。逻辑映射在原本为马尔可夫的混沌系统中分离出表示诱导的记忆，使用其精确不变律作为遍历基准，并利用其零质量伪谱区分弛豫与瞬态放大。精确旋转周期将伪谱校准为鲁棒性诊断而非闭合证书。库存示例给出了闭合驱动的富集过程：停留年龄风险触发年龄细化状态，从而提高预测得分。这些结果催生了最小充分表示：满足预测、结构和可识别性要求的最简化表示。
英文摘要
This work develops a discrete-time framework for identifying probability localization dynamics through finite stochastic representations adapted in space, time, memory, and state information. A compact dynamically relevant set is localized by a finite measurable partition, producing an observable probability state and a relational graph of admissible transitions. Structured stochastic liftings derived from Stochastically Structured Reservoir Computing (SSRC) give lossless polynomial representations of the observable state, while stochastic delay liftings add finite observable memory. These are distinguished from dynamically informed state-space enrichment: refinement of observational fibers containing states with the same present observation but different observable futures, yielding an exact obstruction-to-closure criterion. A route-network toy problem gives a minimal obstruction example, while four numerical laboratories (rotational phase dynamics, the chaotic logistic map, the Van der Pol oscillator, and a synthetic cyclic inventory system) show how spatial scale, temporal scale, polynomial degree, and delay depth interact. The logistic map isolates representation-induced memory in an otherwise Markovian chaotic system, using its exact invariant law as an ergodic benchmark and its zero-mass pseudospectrum to separate relaxation from transient amplification. An exact rotational cycle calibrates pseudospectra as a robustness diagnostic rather than a closure certificate. The inventory example gives a closure-driven enrichment procedure: residence-age hazards trigger age-refined states that improve predictive scores. These results motivate a minimal adequate representation: the least complex representation meeting predictive, structural, and identifiability requirements.
