Beyond RMSE: Structural validation of physics-informed neural network reconstructions via persistent homology

Standard validation metrics such as the root-mean-square error (RMSE) are structurally blind: they measure pointwise accuracy but cannot verify whether a reconstruction preserves the correct number and arrangement of physical features. We present a lightweight computational validation algorithm that uses persistent homology to detect these structural hallucinations in physics-informed neural network (PINN) reconstructions and classical interpolation methods. We demonstrate this topological blindness on three fluid dynamics benchmarks: the viscous Burgers shock, the lid-driven cavity (Re = 100), and a Von Karman vortex street. These examples show that low RMSE does not guarantee structural correctness. Radial basis function (RBF) interpolation achieves a competitive RMSE of 3.6% on the cylinder wake, yet hallucinates 12 persistent vortical structures where only 5 exist. In contrast, PINNs reliably recover the correct macroscopic topology across six independent random initializations on two hardware platforms (H_0 in {3, 5, 6}), making conservative errors rather than hallucinatory ones. We term this the Topological Shield effect: the PDE residual acts as a structural regularizer that suppresses non-physical artifacts. The proposed validation pipeline, built on GUDHI for cubical persistent homology and the Wasserstein-2 distance, scales linearly to evaluate production-scale grids (4096 x 4096) in seconds post-hoc, requires no modification to the reconstruction code, and is applicable to any scalar field produced by any numerical method.