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

Reconstructing hidden physical states from limited measurements is a fundamental task in scientific machine learning for state monitoring and physical analysis. In partially observed partial differential equation (PDE) systems, learned models can exploit statistical correlations in training data to accurately predict state directions that are weakly constrained by observations. Aggregate reconstruction error therefore provides an incomplete picture of how predictive performance relates to observational information. We introduce UniPDE-Bench, a direction-wise evaluation protocol that uses local observation geometry as a reference independent of the estimator. In a common directional basis, the protocol separately assesses empirical recovery, agreement between prediction claims and observation geometry, and confidence-based selection across claim coverages. It constructs a complete right singular basis of the joint observation Jacobian after noise whitening and normalization by a state metric, partitions directions using a relative sensitivity threshold, and evaluates selection through recovery and abstention curves. This geometric reference complements predictive accuracy by characterizing the local observational constraints on selected directions. Experiments across models and observation configurations show that similar aggregate reconstruction errors can correspond to substantially different directional recovery and selection outcomes. In the transport configuration studied, rankings based on conformal interval width achieve an overall geometric ranking score above chance, yet their abstention rate on below-threshold directions falls below the random baseline at some high claim coverages. These findings motivate separate assessment of aggregate predictive accuracy, overall ranking quality, and direction selection at specific coverages. By relating these assessments to local observational constraints, UniPDE-Bench reveals differences in recovery and selection that aggregate metrics obscure.