Limits of Resolution Equivariance in Fourier Neural Operators
For practitioners using FNOs, this work highlights a simple but strong baseline for cross-resolution evaluation and reveals limitations in resolution equivariance that were previously assumed.
Fourier Neural Operators (FNOs) are often assumed to generalize across resolutions, but this work shows that direct fine-grid inference is not reliably beneficial and can be worse than a simple low-grid-plus-upsampling baseline on Darcy flow. The study identifies nonlinear aliasing as a key obstacle to zero-shot resolution equivariance.
Fourier Neural Operators are often assumed to generalize across spatial resolutions, enabling training on a coarse grid and deployment on a finer grid. We test this assumption by contrasting two inference-time choices when moving from training resolution $s$ to test resolution $S>s$: running FNO directly at $S$, or running at $s$ and upsampling the prediction to $S$ via Fourier zero-padding. On Darcy flow, we observe that direct fine-grid inference is not reliably beneficial and can be worse than the low-grid-plus-upsampling baseline. We further analyze layerwise spectra and find that, under Fourier truncation, intermediate representations increasingly concentrate energy in low frequencies, with high-frequency output produced mainly by late nonlinear/decoder stages. This offers a mechanistic explanation for why FNO can perform well while retaining few modes, yet remain sensitive under resolution shifts. Our findings highlight a simple but strong baseline for cross-resolution evaluation and point to nonlinear aliasing as a key obstacle to zero-shot resolution equivariance.