Collocation geometry creates structural blind spots in physics-informed neural networks

Abstract

Physics-informed neural networks (PINNs) minimize empirical residuals of governing equations, but a small sampled residual need not certify the structural properties of the underlying partial differential equation (PDE). We examine this issue from a discrete least-squares viewpoint using sampled-residual observability as a diagnostic lens. The central analytic contribution is an explicit nontrivial null direction of the sampled strong-form Poisson residual on a structured tensor-product grid, yielding an affine family of residual-indistinguishable fields. We then examine three diagnostic settings. In hyperbolic transport, global-in-time residual minimization produces a weak pre-echo ahead of the transported front. In a parabolic battery-inspired surrogate, residual-only and anchored PINNs violate expected nonnegativity unless boundary-compatible or positivity-preserving structure is encoded. In a heated-pipe boiling reconstruction, a drift-flux model-informed PINN can match digitized void-fraction data while violating 0 ≤ α ≤ 1 unless admissibility is built into the representation. Controlled tests connect these effects to observability. In an elliptic persistence test, structured collocation retains an initialized blind component with a/a0 = 1.00 ± 0.00, whereas a 1% jittered grid and an equal-count random cloud drive it to numerical zero. In the battery surrogate, hard-boundary and hard-positive parameterizations eliminate all observed violations across five seeds, while a nested zero-noise anchor audit shows that increasing midline anchors from 10 to 60 does not provide a structural admissibility guarantee. The contribution is diagnostic, rather than prescriptive: low sampled residual should not be treated as a certificate of PDE structure unless that structure is made observable or encoded directly.