FNO在不同分辨率下表现不稳定，直接细粒度推理未必更优。
Limits of Resolution Equivariance in Fourier Neural Operators
- 对比直接细网格推理与粗网格上采样，发现后者更可靠。
- 中间层频谱能量集中于低频，高频信息主要由后期非线性模块生成。
- 揭示了FNO对分辨率变化敏感的机制，适合关注模型泛化性的研究者。
傅里叶神经算子常被假设具备跨空间分辨率的泛化能力，即在粗网格上训练后可部署于更细网格。我们通过对比从训练分辨率 $s$ 到测试分辨率 $S>s$ 的两种推理方式来检验该假设：直接在 $S$ 上运行FNO，或在 $s$ 上运行并用傅里叶零填充上采样至 $S$。在达西流任务中，直接细网格推理并不总优于粗网格加上采样的基线，甚至可能更差。进一步分析层间频谱发现，在傅里叶截断下，中间表示的能量逐渐集中在低频，高频输出主要由后期非线性/解码器阶段产生。这为FNO虽保留少量模式却仍能表现良好、但在分辨率变化下敏感提供了机制解释。研究强调了一个简单而强大的跨分辨率评估基线，并指出非线性混叠是实现零样本分辨率等变性的关键障碍。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。