Identifiability before Optimization: Sparse Observation Design for PDE Search

A vanishing physics residual can coexist with large solution error when the exposed partial differential equation (PDE) specification leaves the target ambiguous. We isolate this information failure by removing a public condition while keeping the hidden target fixed. For finite-dimensional linear operators, sparse observations restore identifiability exactly when they have full rank on the operator's nullspace; their conditioning controls reconstruction stability. We extend the analysis to estimated ambiguity spaces and local nonlinear problems. Motivated by these conditions, a target-independent portfolio selects observation locations before querying their values. Across seven controlled symbolic-search families, omitting initial data yields a perfect proxy optimum with relative L2 error of 1.000; nine public observations reduce mean error to 0.088, versus 0.074 with the full condition. On 16 PDE libraries, using ten observations with 5% representation mismatch and 1% Gaussian noise, the portfolio achieves mean error of 0.0587, improves on uniform placement in 15/16 tasks, and meets an absolute 0.01 non-inferiority margin to A-optimal design. Independent neural-solver experiments reproduce the omission–repair gap: eight observations improve all 24 paired DeepXDE runs, although Allen–Cahn remains inaccurate. These results distinguish information sufficiency from optimization success and show how ambiguity analysis can guide sparse observation design.