Spectral Reduction of Failure Modes in Physics-Informed Neural Networks

A persistent assumption in the physics-informed neural network (PINN) literature is that training failures stem from extreme ill-conditioning of the Neural Tangent Kernel (NTK). We test this assumption by conducting a spectral audit across a subset of the Atlas of PINN Failures. Out of 29 total experiments, we analyze n=20 checkpoints, excluding 9 due to architecture mismatch or missing weights. We find that the full NTK condition number k_full fails as a diagnostic metric within this dataset. Both successful controls and catastrophic failures universally exhibit condition numbers exceeding 10^15. Instead, our results suggest that PINN failures organize into a three-tier convergence structure within the evaluated dataset. The Burden Transfer Principle (Tier 1) and Fourier-representation mismatch (Tier 2) operate through distinct mechanisms but convergently manifest as a block-wise eigenvalue imbalance, where the PDE block dominates boundary conditions by ratios reaching 35,061. Conversely, structural collapse (Tier 3) maintains a balanced block ratio of 9.86 but yields a rank-deficient NTK. These observations suggest that, in the cases examined here, training failure reduces to a terminal block-wise imbalance or rank deficiency rather than global ill-conditioning.