# Limits of Resolution Equivariance in Fourier Neural Operators

arXiv: https://arxiv.org/abs/2606.00677 · HTML: https://mlsanity.com/paper/2606.00677/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 7 references that are in the map.
- First posted on arXiv 30 May 2026.

## Builds on (most cited first, 3 of 3)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Neural Operators Explained: A Practical Perspective](https://mlsanity.com/paper/2512.01421/) (2025)
- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)

## Abstract (arXiv)

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.