Limits of Resolution Equivariance in Fourier Neural Operators

Alexandre Allauzen; Alex Colagrande; Eva Feillet; Paul Caillon

arxiv: 2606.00677 · v1 · pith:5JNBTM2Gnew · submitted 2026-05-30 · 💻 cs.LG

Limits of Resolution Equivariance in Fourier Neural Operators

Alex Colagrande , Paul Caillon , Eva Feillet , Alexandre Allauzen This is my paper

Pith reviewed 2026-06-28 19:26 UTC · model grok-4.3

classification 💻 cs.LG

keywords Fourier Neural Operatorsresolution equivarianceDarcy flowspectral analysisnonlinear aliasingneural operatorsgeneralizationFourier truncation

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The pith

Fourier Neural Operators show limited resolution equivariance because direct fine-grid inference does not reliably outperform low-grid inference followed by upsampling.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests the assumption that Fourier Neural Operators trained on coarse grids can be deployed directly on finer grids. It contrasts two inference strategies when moving from training resolution s to test resolution S: running the model directly at S versus running at s and upsampling the output via Fourier zero-padding. On Darcy flow, direct fine-grid inference is not reliably better and can be worse than the upsampling baseline. Layerwise spectral analysis shows that intermediate representations concentrate energy in low frequencies under truncation, with high frequencies emerging mainly from late nonlinear stages. This supplies a mechanistic account for why FNOs succeed with few modes yet remain sensitive to resolution changes, identifying nonlinear aliasing as the obstacle to zero-shot equivariance.

Core claim

On Darcy flow, direct fine-grid inference is not reliably beneficial and can be worse than the low-grid-plus-upsampling baseline. 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.

What carries the argument

The head-to-head comparison of direct high-resolution FNO inference against low-resolution inference plus Fourier zero-padding upsampling, together with layerwise spectral energy analysis.

If this is right

A low-resolution inference plus Fourier upsampling baseline should be included in any cross-resolution FNO evaluation.
Nonlinear aliasing prevents true zero-shot resolution equivariance in FNOs.
FNO success with few retained modes stems from progressive low-frequency concentration across layers.
High-frequency output details depend primarily on the final nonlinear and decoder stages.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same early-layer low-frequency concentration pattern may appear in other spectral operator architectures and limit their cross-resolution behavior.
Regularization or architectural changes that reduce nonlinear aliasing could improve resolution generalization without changing the number of modes.
Repeating the layerwise spectrum analysis on Navier-Stokes or other benchmark datasets would test whether the Darcy flow findings are dataset-specific.
The interaction between chosen truncation levels and the underlying PDE physics may drive part of the observed sensitivity.

Load-bearing premise

The Darcy flow dataset and chosen truncation levels are representative of FNO behavior under resolution shifts in general.

What would settle it

An experiment in which direct fine-grid inference consistently and substantially outperforms the low-grid-plus-upsampling baseline across multiple resolutions and additional PDE datasets would falsify the central observation.

Figures

Figures reproduced from arXiv: 2606.00677 by Alexandre Allauzen, Alex Colagrande, Eva Feillet, Paul Caillon.

**Figure 1.** Figure 1: Cross-resolution inference vs. a band-limited upsampling baseline (s = 85 → S = 211). Top: ground-truth at s = 85, Direct-S prediction at S = 211, and Low+pad prediction (run at s then upsample the output to S via Fourier zero-padding), shown with no spatial padding and spatial padding = 10. Bottom: corresponding radially averaged power spectra. Direct evaluation at S produces a spurious high-frequency tai… view at source ↗

**Figure 2.** Figure 2: Layerwise radially averaged power spectra under different Fourier truncation levels. From left to right: initialization (no truncation), then truncation to 4 × 4, 8 × 8, and 64 × 64 modes. Curves show the radially averaged power spectrum of the input, intermediate feature maps (after lifting and successive layers), and final output. The vertical red dashed line indicates the maximum retained Fourier mode… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Direct fine-grid FNO inference underperforms upsampling on Darcy flow, but the nonlinear aliasing explanation lacks supporting controls.

read the letter

The main takeaway is that on Darcy flow, running an FNO directly at fine test resolution can be no better or even worse than training at coarse resolution and upsampling the output with Fourier zero-padding. The paper also shows layerwise spectra where intermediate layers concentrate energy at low frequencies and high frequencies appear mostly in the final nonlinear stages.

The explicit head-to-head test of those two inference choices is a straightforward addition that prior FNO work has not emphasized. The spectra give a plausible account of why the model can stay accurate with heavy truncation yet still feel resolution shifts. That part is grounded in the standard FNO architecture and does not invent new entities.

The soft spot is the jump from these observations to nonlinear aliasing as a key general obstacle. The spectra are descriptive for the chosen truncation and this dataset; there are no ablations that vary high-frequency content while holding other factors fixed, and no results on other PDEs. Without those, the performance gap could trace to Darcy-specific statistics or training dynamics rather than a fundamental limit on resolution equivariance. The abstract also omits error bars or run counts, which makes the reliability of the "not reliably beneficial" claim harder to judge.

This is useful for groups that train neural operators on one grid and deploy on another in scientific computing. It deserves a serious referee because the empirical comparison is simple to check and the practical question is live, even though the mechanistic claim will need tighter evidence in revision.

Referee Report

2 major / 1 minor

Summary. The paper claims that FNOs do not reliably benefit from direct fine-grid inference when moving from training resolution s to test resolution S > s on Darcy flow; instead, low-grid inference followed by Fourier zero-padding upsampling can perform as well or better. Layerwise spectral analysis is used to argue that intermediate layers concentrate energy in low frequencies under truncation, with high frequencies generated mainly in late nonlinear stages, providing a mechanistic account for why FNOs remain sensitive to resolution shifts despite retaining few modes. The work positions nonlinear aliasing as a key obstacle to zero-shot resolution equivariance and recommends the upsampling baseline for cross-resolution evaluation.

Significance. If the empirical observation and spectral analysis hold under proper statistical controls, the result supplies a simple, strong baseline for resolution-shift evaluation in neural operators and identifies a concrete limitation that could guide architecture or training modifications for PDE surrogate modeling. The absence of parameter fitting or circular derivations is a strength of the empirical framing.

major comments (2)

[paragraph on experimental setup] Paragraph on experimental setup (as referenced in the abstract): the claim that direct fine-grid inference 'is not reliably beneficial and can be worse' is presented without error bars, dataset sizes, number of runs, or statistical tests. This makes it impossible to determine whether observed differences exceed variability or arise from post-hoc truncation choices.
[Abstract / layerwise spectra analysis] Abstract and layerwise spectra discussion: the step from the Darcy-specific performance gap to the general claim that 'nonlinear aliasing [is] a key obstacle to zero-shot resolution equivariance' lacks controls (e.g., varying truncation levels or high-frequency content to modulate the direct-vs-upsample gap) or results on additional PDEs. Without such evidence the causal link between the observed spectra and the resolution-shift behavior remains unsecured.

minor comments (1)

The abstract would be clearer if it stated the specific truncation levels (number of modes) and grid sizes used for the Darcy experiments.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive feedback. We address each major comment below and will revise the manuscript to improve statistical reporting and strengthen the evidential basis for our claims where feasible.

read point-by-point responses

Referee: Paragraph on experimental setup (as referenced in the abstract): the claim that direct fine-grid inference 'is not reliably beneficial and can be worse' is presented without error bars, dataset sizes, number of runs, or statistical tests. This makes it impossible to determine whether observed differences exceed variability or arise from post-hoc truncation choices.

Authors: We agree that explicit reporting of variability is necessary. The experiments are performed on the standard Darcy flow dataset (1000 training samples, 200 test samples). In the revision we will add error bars from 5 independent runs with different random seeds, report the number of runs explicitly, and confirm that the performance gaps between direct fine-grid inference and the upsampling baseline remain consistent across seeds. revision: yes
Referee: Abstract and layerwise spectra discussion: the step from the Darcy-specific performance gap to the general claim that 'nonlinear aliasing [is] a key obstacle to zero-shot resolution equivariance' lacks controls (e.g., varying truncation levels or high-frequency content to modulate the direct-vs-upsample gap) or results on additional PDEs. Without such evidence the causal link between the observed spectra and the resolution-shift behavior remains unsecured.

Authors: The layerwise spectra are presented as a mechanistic explanation for the Darcy results under Fourier truncation. We will add new experiments that vary the number of retained modes to demonstrate how the direct-vs-upsample gap changes with truncation level, thereby tightening the connection between the spectral observations and the resolution-shift behavior. The abstract claim is scoped to FNOs; we will qualify the wording to reflect that the evidence is drawn from this architecture and benchmark. revision: partial

standing simulated objections not resolved

Results on PDEs other than Darcy flow.

Circularity Check

0 steps flagged

No circularity: purely empirical comparison and spectral observation

full rationale

The paper reports experimental results on Darcy flow (direct fine-grid inference vs. low-resolution plus upsampling) and layerwise Fourier spectra under truncation. No derivation chain, fitted parameters renamed as predictions, self-definitional equations, or load-bearing self-citations appear in the provided text. Claims rest on observed performance differences and energy concentration patterns rather than any reduction of outputs to inputs by construction. The generalization to 'nonlinear aliasing as obstacle' is an interpretive step, not a tautological one.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The claim rests entirely on empirical observations from one dataset; no mathematical free parameters, axioms, or invented entities are introduced.

pith-pipeline@v0.9.1-grok · 5703 in / 974 out tokens · 22268 ms · 2026-06-28T19:26:02.988928+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 2 canonical work pages

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Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning , author=. arXiv preprint arXiv:2506.10973 , year=

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