Auditing Coefficient Identifiability in Physics-Constrained Dictionary Models

Low data-fitting error and a small physics residual do not ensure that the coefficients of a dictionary-based PDE solution are identifiable. For a fixed linear operator and sampling design, we develop a pre-training structural audit based on the classical joint null space of the data and residual maps. Without observed response values, the audit identifies coefficient perturbations that leave both predictions and residuals unchanged, returns explicit witnesses, and reports which directions the physics penalty constrains. Tolerance scans, higher-precision calculations, and symbolic checks distinguish functional identities from sampling-induced dependencies and numerical near-dependencies. Across nineteen dictionaries, the audit detects multi-term dependencies missed by pairwise and triple screens and identifies four dictionaries with identically zero residual penalties. In a public PySINDy constrained-regression example, steady-state observations leave seven coefficient directions unresolved by the data and constraint together. On these data, the published optimizer fits the sampled derivatives with a relative ${\ell }_2$ error of $1.7\times 10^{-5}$ while omitting the growth and coupling terms. Quadrupling the steady-state samples leaves all seven directions unresolved. At the original sample budget, transient observations remove these blind directions. The audit identifies which individual coefficients and coefficient combinations are determined under the specified observations and physics constraints. It does not validate the imposed operator or guarantee improved reconstruction accuracy.