When Lower Error Misses the Physics. Parametric Generalisation of Neural PDE Surrogates

Learned PDE surrogates are increasingly deployed across varying physical regimes, yet their out-of-distribution (OOD) generalisation remains poorly understood. This is particularly critical in turbulent flows, where shifts in governing parameters fundamentally alter the solution's structural scales. We study parametric generalisation on the \texttt{shear\_flow} dataset using controlled interpolation and extrapolation tests along the Reynolds ($Re$) and Schmidt ($Sc$) axes. Comparing a Cross-Shaped Window Transformer (CSWin) and a Factorised Fourier Neural Operator (FFNO), we find that unconditioned models can lose parameter-dependent spectral structure under extrapolation. Explicit conditioning changes this behaviour, but its effect depends strongly on architecture and on how generalisation is evaluated. For CSWin, dynamic FiLM conditioning drastically reduces OOD rollout error compared to conditioning with static spatial channels, yet spectral diagnostics show both methods recover the parameter-dependent power spectra similarly well. Conversely, for FFNO, spatial conditioning can actively degrade nominal VRMSE while simultaneously improving the physical separation of spatial spectra. These results demonstrate that a surrogate's pointwise accuracy can diverge from recovery of parameter-dependent physical structure. We argue that evaluating parametric generalisation requires pairing aggregate prediction error with additional diagnostics (e.g. spectral rollout) to ensure models learn the correct response to changing physics.