When Does a Fourier Neural Operator Fail? Spectral Diagnostics for Low-Rank Surrogates of Reaction–Diffusion

Neural surrogates for PDE simulators are usually judged by one-step error, which hides the two failures that matter in practice: error accumulation under autoregressive rollout and collapse on unseen parameter regimes. We study a compact CP-factorised Fourier neural operator (LiteFNO) against a parameter-matched CNN on the Gray-Scott benchmark from The Well, asking not only whether the surrogate fails but why. (i) LiteFNO needs 22% fewer FLOPs than the CNN, but both its one-step edge and its rollout advantage in a single seed reverse under three-seed averaging; rollout error is instead governed by the fate of the high-frequency spectrum, which the CNN over-damps and LiteFNO over-excites. (ii) Causally ablating the lowest-magnitude retained Fourier modes shows they contribute little at one step yet dominate mid-horizon rollout error. (iii) In a preliminary leave-one-regime-out study, failure on a held-out Gray-Scott regime correlates with the share of simulator spatial variance at low wavenumber (Spearman $\rho=0.94$ over six regimes, single seed), a statistic requiring no trained model. We also report noise and 6-bit quantisation robustness, CPU/GPU latency, and a deferral pilot in which ensemble disagreement captures essentially all of the oracle-achievable gain, while showing that out-of-distribution error is spread across steps rather than concentrated in a discardable minority. These results argue for spectral diagnostics, computed from simulator output alone, as a cheap pre-training predictor of surrogate reliability.