Title: Examining the Boundaries of Resolution Equivariance in Fourier Neural Operators

Abstract: A common premise regarding Fourier Neural Operators (FNOs) is their ability to generalize across varying spatial resolutions, allowing models to be trained on coarse grids and subsequently deployed on finer ones. This study scrutinizes that assumption by comparing two distinct inference strategies when transitioning from a training resolution $s$ to a higher test resolution $S>s$: executing the FNO directly at resolution $S$ versus running the model at resolution $s$ and subsequently upsampling the output to $S$ using Fourier zero-padding. Our experiments on Darcy flow reveal that direct inference at the finer grid does not consistently yield improvements and may, in fact, underperform the baseline approach of upsampling after low-resolution prediction. Furthermore, an analysis of layerwise spectra demonstrates that when Fourier truncation is applied, intermediate feature representations progressively concentrate their energy in lower frequencies. Consequently, high-frequency details in the final output are primarily generated by the late-stage nonlinear transformations and the decoder. This mechanistic insight explains why FNOs can achieve strong performance with a limited number of modes while simultaneously remaining vulnerable to resolution shifts. These results underscore the importance of a straightforward yet robust baseline for cross-resolution assessments and identify nonlinear aliasing as a primary barrier to achieving zero-shot resolution equivariance.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC