发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences; AMSS Center for Forecasting Science, Chinese Academy of Sciences; State Key Laboratory of Mathematical Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; Beijing Institute of Mathematical Sciences and Applications (BIMSA); Department of Statistics and Data Science, Tsinghua University; Paula and Gregory Chow Institute for the Studies in Economics, Xiamen University; Department of Statistics, London School of Economics and Political Science; Yau Mathematical Sciences Center, Tsinghua University（中国科学院数学与系统科学研究院; 中国科学院数学与系统科学研究院预测科学中心; 中国科学院数学与系统科学研究院数学科学学院国家重点实验室; 北京国际数学研究中心; 清华大学统计与数据科学系; 厦门大学保罗和格雷戈里周经济研究所; 伦敦政治经济学院统计学系; 清华大学丘成桐数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对非线性滤波中无限李代数导数爆炸导致的滤波发散问题，提出基于统计纤维理论的SMG-Yau-Yau滤波器，通过正交分解隔离非线性并引入AAT泛函与规范对称性破缺机制，实现无损数据同化与稳定收敛。
AI 中文摘要
经典连续时间非线性滤波在一般非线性状态空间中因无限李代数导数爆炸（$\dim(\mathcal{E}) = \infty$）而失效，导致滤波发散。为解决这一持续四十年的危机，我们提出了SMG-Yau-Yau滤波器和基于Orlicz流形$\mathcal{M}$上的统计纤维理论的无损数据同化框架。通过为$\mathcal{M}$配备黎曼淹没和Ehresmann联络，无约束的Duncan-Mortensen-Zakai得分速度场被正交分解为统计可验证方向（$\text{SVD}χ_f$）和结构内部方向（$\text{SID}_f$）。系统非线性和未闭合的李交换子被正交隔离在$\text{SID}_f$中，保护宏观基础参数免受空间导数污染，同时保留总得分方差能量。我们通过双轴坍缩机制统一了渐近性：证明了当$\dim(\mathcal{E}) < \infty$时，我们的模型恒等地恢复经典Yau-Yau动力学，而大样本极限（$N \to \infty$）在$\dim(\mathcal{E}) = \infty$下诱导热力学淬灭，使无限维几何平坦化。最后，我们提出了主动非因果张力（AAT）泛函，用于在线监测累积的模型误设。超过拓扑容量阈值触发规范对称性破缺（GSB），动态扩展基础坐标（$d \to d+1$），以确保非渐近稳定性和对精确状态密度的收敛。
英文摘要
Classical continuous-time non-linear filtering fails in generic non-linear state spaces due to an infinite Lie algebraic derivative explosion ($\dim(\mathcal{E}) = \infty$), leading to filter divergence. To resolve this four-decade crisis, we introduce the SMG-Yau-Yau filter and lossless data assimilation framework built on Statistical Fiber Theory over an Orlicz manifold $\mathcal{M}$. By equipping $\mathcal{M}$ with a Riemannian submersion and an Ehresmann connection, the unconstrained Duncan-Mortensen-Zakai score velocity field is orthogonally decomposed into Statistically Verifiable Directions ($\text{SVD}χ_f$) and Structural Internal Directions ($\text{SID}_f$). System non-linearities and unclosed Lie commutators are orthogonally quarantined in $\text{SID}_f$, protecting macroscopic base parameters from spatial derivative pollution while preserving total score variance energy. We unify the asymptotics through a Dual-Axis Collapse Mechanism: proving our model identically recovers classical Yau-Yau dynamics when $\dim(\mathcal{E}) < \infty$, while large-sample limits ($N \to \infty$) induce a thermodynamic quench that flattens infinite-dimensional geometry under $\dim(\mathcal{E}) = \infty$. Finally, we formulate the Active Acausal Tension (AAT) functional to monitor accumulated model misspecification online. Exceeding a topological capacity threshold triggers Gauge Symmetry Breaking (GSB), which dynamically expands base coordinates ($d \to d+1$) to ensure non-asymptotic stability and convergence to the exact state density.
