Recoverability Is a Subspace Property: A Benchmark for Certified State Estimation from Partial PDE Observations

2026-09-11

AI Summary

研究通过引入UniPDE-Bench协议，利用局部观测几何评估基于偏微分方程系统的状态估计，解决了传统方法无法有效衡量每个预测方向约束强度的问题。

Abstract

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.

Problem

• partial differential equation
• state estimation
• observation geometry

Innovation

• UniPDE-Bench
• direction-wise evaluation
• local observation geometry
• state estimation
• partial PDE observations