可恢复性是子空间性质：基于偏微分方程部分观测的认证状态估计基准
Recoverability Is a Subspace Property: A Benchmark for Certified State Estimation from Partial PDE Observations
浏览论文内容
中文总结 AI 辅助
针对PDE部分观测的状态估计，提出方向性评估协议UniPDE-Bench，通过白化雅可比矩阵划分方向，独立于估计器评估恢复与置信选择，揭示聚合误差掩盖的局部灵敏度问题。
中文摘要 AI 辅助
当从部分观测中重建偏微分方程（PDE）系统的隐藏状态时，聚合预测误差衡量的是在给定数据分布上的性能，但并未揭示观测对每个预测方向的约束强度。我们引入了UniPDE-Bench，一种方向性评估协议，它将局部观测几何结构纳入状态估计评估中，为预测恢复和基于置信度的选择提供了独立于估计器的参考。该协议通过噪声协方差对联合观测雅可比矩阵进行白化，并用状态度量对其进行归一化。其完整的右奇异基表示状态的联合变化，相对灵敏度阈值将其划分为保留方向和低于阈值方向。在此共同基下，该协议评估恢复能力以及预测声明与几何划分之间的一致性，而恢复曲线和弃权（不执行）曲线描述了在不同声明覆盖率下基于置信度的选择。在我们研究的模拟任务和观测配置中，总体排名优于随机水平的置信规则在某些高声明覆盖率下，对低于阈值方向的弃权（不执行）仍可能低于随机水平。通过将经验恢复与方向选择质量分开，该协议将预测性能与局部观测灵敏度联系起来，并为部分观测状态估计器提供了超越聚合误差的评估。
英文摘要
When reconstructing the hidden state of a partial differential equation (PDE) system from partial observations, aggregate prediction error measures performance on a given data distribution but does not reveal how strongly the observations constrain each predicted direction. We introduce UniPDE-Bench, a direction-wise evaluation protocol that incorporates local observation geometry into state-estimation assessment, providing a reference for prediction recovery and confidence-based selection that is independent of the estimator. The protocol whitens the joint observation Jacobian by the noise covariance and normalizes it by a state metric. Its complete right singular basis represents joint variations of the state, which a relative sensitivity threshold partitions into retained and below-threshold directions. In this common basis, the protocol evaluates recovery and the agreement between prediction claims and the geometric partition, while recovery and abstention curves describe confidence-based selection at different claim coverages. In the simulated tasks and observation configurations studied here, confidence rules whose overall ranking exceeds chance can still exhibit below-random abstention on below-threshold directions at some high claim coverages. By separating empirical recovery from direction-selection quality, the protocol relates prediction performance to local observation sensitivity and provides an evaluation of partially observed state estimators beyond aggregate error.
发表机构
- Key Laboratory of Networked Control Systems, Chinese Academy of Sciences（中国科学院网络化控制系统重点实验室）
- University of Chinese Academy of Sciences（中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。
