AI 中文总结
本研究提出动力学不确定性几何（DUG）框架，区分后验不确定性、未来观测与模型输运三要素，通过质量索引持续同调模建立稳定性结果，经定轨与三体问题基准验证，可更丰富描述不确定性。
AI 中文摘要
非线性动力学系统中的不确定性常由仅从后验几何中无法明显看出的输运结构所组织。我们引入动力学不确定性几何（DUG）这一框架，将三个常被混淆的要素区分开来：后验不确定性、未来观测以及模型诱导的输运。DUG通过共同的 enclosed-probability 坐标对后验可信集和输运条件子集进行索引，从而能在不同可信度水平上追踪具有物理意义的类别，而非仅在单一阈值下进行考察。该框架结合了质量排序的可信滤波、通过未来实验诱导的概率定律与费舍尔几何对未来实验的描述，以及代表动力学上不同结果的标记输运骨架。我们在同时对后验密度和概率测度进行扰动的情况下，为所得的质量索引持续同调模建立了稳定性结果，包括连续统、细化和有限原子形式。这些结果为从数值近似和加权网格计算得到的持续性提供了定量控制。两个基准问题对该框架进行了说明：在短弧定轨场景中，DUG在连通的不确定性区域内识别出动力学上不同的返回类别，并凸显了基于信息的观测设计准则与基于局部费舍尔的观测设计准则之间的差异；在地月平面圆形限制性三体问题中，DUG揭示了连通可信区域内共存的多个首次撞击输运结果，并为评估其数值分辨率提供了诊断工具。总体而言，这些示例表明，后验几何的拓扑摘要与输运标记和未来实验相结合，能比单独的后验概率或动力学分类产生更丰富的不确定性描述。
英文摘要
Uncertainty in nonlinear dynamical systems is often organized by transport structures that are not apparent from posterior geometry alone. We introduce Dynamical Uncertainty Geometry (DUG), a framework that separates three ingredients that are frequently conflated: posterior uncertainty, future observations, and model-induced transport. DUG indexes posterior credible sets and transport-conditioned subsets by a common enclosed-probability coordinate, allowing physically meaningful classes to be tracked across credibility levels rather than examined at a single threshold. The framework combines a mass-ranked credible filtration, a description of future experiments through their induced probability laws and Fisher geometry, and a labelled transport skeleton representing dynamically distinct outcomes. We establish stability results for the resulting mass-indexed persistence modules under simultaneous perturbations of posterior density and probability measure, including continuum, refinement, and finite atomic formulations. These results provide quantitative control of persistence computed from numerical approximations and weighted grids. Two benchmark problems illustrate the framework. In a short-arc orbit-determination setting, DUG identifies dynamically distinct return classes within a connected uncertainty region and highlights a difference between information-based and local Fisher-based observation-design criteria. In the Earth-Moon planar circular restricted three-body problem, DUG reveals multiple first-hit transport outcomes coexisting within a connected credible region and provides diagnostics for assessing their numerical resolution. Together, these examples show how topological summaries of posterior geometry, when coupled to transport labels and future experiments, yield a richer description of uncertainty than either posterior probabilities or dynamical classifications alone.
