Limits of Resolution Equivariance in Fourier Neural Operators

2026-05-30

AI Summary

This study investigates the practical performance of Fourier Neural Operators (FNOs) in cross-resolution generalization and critically examines the validity of their assumed resolution equivariance. By comparing direct high-resolution inference against low-resolution inference followed by Fourier zero-padding upsampling, and complemented with inter-layer spectral analysis, the work reveals that FNOs concentrate spectral energy toward low frequencies in intermediate layers, while high-frequency details are predominantly reconstructed in later nonlinear stages. The authors identify nonlinear aliasing as a key factor impeding zero-shot resolution equivariance. Notably, on the Darcy flow benchmark, direct high-resolution inference does not consistently outperform the upsampling baseline, prompting the proposal of a simple yet effective cross-resolution evaluation protocol.

Abstract

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.

Problem

• resolution equivariance
• Fourier Neural Operators
• cross-resolution generalization
• nonlinear aliasing
• spectral concentration

Innovation

• Fourier Neural Operators
• resolution equivariance
• nonlinear aliasing
• spectral analysis
• zero-shot generalization