AI 中文总结
本书将几何方法用于随机动力系统分析，把转移重述为Schrödinger桥，通过α-散度推广并引入信息测地线，为复杂系统在不确定性下跨亚稳态移动提供统一几何阐释。
AI 中文摘要
几何方法对于分析、预测和缓解非线性系统固有的复杂行为不可或缺。在该领域中，最概然转移路径使Onsager-Machlup作用泛函最小化，标记了跨能量势垒的最可能路径。本书将分析从单个样本路径提升至概率密度的无限维空间，把这些转移重述为Schrödinger桥——通过最小化相对熵定义的边界分布间的最优路径，并证明当亚稳态被理想化为狄拉克δ函数时，Onsager-Machlup路径会作为特例出现。进一步通过α-散度进行推广，α-散度与熵及非平衡转移的热力学成本相关，本书引入信息测地线作为由此产生的最优密度路径，为复杂系统在不确定性下于亚稳态间的移动提供了统一的几何阐释。
英文摘要
Geometric methods are indispensable for analyzing, predicting, and mitigating the complex behaviors inherent in nonlinear systems. In this regime, the most probable transition path minimizes the Onsager-Machlup action functional, marking the likeliest route across an energy barrier. Lifting the analysis from individual sample paths to the infinite-dimensional space of probability densities, the book recasts these transitions as Schrödinger bridges - optimal paths between boundary distributions defined by minimizing relative entropy - and shows that the Onsager-Machlup path emerges as a special case when metastable states are idealized as Dirac masses. Generalizing further through α-divergences, which connect to entropies and the thermodynamic cost of nonequilibrium transitions, it introduces information geodesics as the resulting optimal density paths, offering a unified geometric account of how complex systems move between metastable regimes under uncertainty.
