What Does a Neural PDE Solver Really Learn? A Residual-Spectrum Diagnostic

Ali Baheri

Keywords: Residual-Spectrum Diagnostics (RSD), physics residual, Fourier / spectral analysis, neural PDE solvers, physics compliance

TL;DR: RSD evaluates neural PDE solvers by FFT-analyzing their PDE residuals and summarizing scale-wise physics violations with HFV/LFV scores.

Abstract: Neural PDE surrogates are typically evaluated using solution error (e.g., relative $L_2$), but low error does not guaranty that predictions satisfy the governing equations. We propose \emph{Residual-Spectrum Diagnostics} (RSD), which evaluates physics compliance by analyzing the spatial frequency content of the PDE residual computed on model rollouts. RSD summarizes scale-dependent violations using two indices: High-Frequency Violation (HFV) for spurious small-scale artifacts and Low-Frequency Violation (LFV) for large-scale dynamical errors. On the 1D viscous Burgers equation, two models with similar relative $L_2$ error differ substantially in residual spectra: training with high-frequency corrupted targets increases $L_2$ error by only $6\%$ but increases HFV by $37\%$ ($p<10^{-3}$). These results show that residual-spectrum analysis reveals physics failures that aggregate error metrics can miss and provides an actionable complement to standard evaluation.

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Submission Number: 25