arXiv:2608.24940cs.LGcs.AI2026-08

拆分高低频成分能显著提升复杂PDE的求解精度,但对简单问题效果有限。

When Does Frequency Decomposition Benefit Physics-Informed Neural Networks? A Preliminary Ablation Study

论文配图:When Does Frequency Decomposition Benefit Physics-Informed Neural Networks? A Preliminary Ablation Study
图 1 · 摘自论文原文
  • 设计双分支谱门架构,分离并动态路由高低频信息
  • 在多尺度波问题上误差降低59.2%,但对平滑问题无效甚至更差
  • 适合高频振荡或多重尺度的物理建模任务,不适用于简单情形

偏微分方程(PDE)常具有高频与多尺度特征,神经网络难以逼近。物理信息神经网络(PINNs)虽将控制方程嵌入训练,却存在频谱偏差:低频成分学习快于高频。傅里叶特征嵌入和正弦激活等方法虽被广泛采用,但多数研究假设其普适有效,未验证具体频段收益。本文提出双分支谱门架构(DBSG-PINN),将高低频成分分别送入子网络,通过自适应门控融合。在五个一维基准PDE上测试,涵盖从光滑单尺度到振荡多尺度的问题。结果显示,频率分解在谱复杂问题上最有效,使多模态波问题相对 $L_2$ 误差下降最多达 $59.2/$;但在平滑问题上几乎无益,甚至在1D Wave上表现劣于固定组合基线。门控机制优势随目标解谱丰富度增加而增强,全模型在多尺度任务中领先,而在单尺度任务中优势最小或为负,表明其利用了频率结构而非噪声。所有结果基于单一训练种子,属探索性研究,旨在引发思考,非定论,需更多种子与基准验证模式是否稳健。

原文摘要 · Abstract (English)

Partial differential equations (PDEs) often have high-frequency and multi-scale features that neural networks struggle to approximate. Physics-Informed Neural Networks (PINNs) build the governing equations directly into training, but suffer from spectral bias: they learn low-frequency components faster than high-frequency ones. Techniques such as Fourier feature embeddings and sinusoidal activations address this, but most studies assume they help across the board without checking which spectral regimes actually benefit. We introduce a dual-branch, spectrally-gated architecture (DBSG-PINN) that splits low- and high-frequency components into separate subnetworks joined by an adaptive gate, and use it to run a partially controlled ablation of frequency decomposition and spectral routing. We test this on five one-dimensional benchmark PDEs, ranging from smooth, single-scale problems to oscillatory, multi-scale ones. Frequency decomposition helps most on the spectrally complex benchmarks, cutting relative $L_2$ error by up to $59.2\%$ on a multimodal wave problem, but gives little benefit on smoother PDEs. On one benchmark (1D Wave), it performs substantially worse than a simpler fixed-combination variant. The gate's benefit scales with how spectrally rich the target solution is: the full model's advantage over the ablations is largest on multi-scale benchmarks and smallest (or negative) on single-scale ones, consistent with the gate exploiting frequency structure rather than acting as noise,though we do not directly visualize or quantify its spatial activations in this study. All results come from a single training seed across five 1D benchmarks, so we present this as an exploratory study meant to raise questions rather than answer them, and outline the additional seeds and benchmarks needed to test whether the pattern holds.

PINN频谱分析多尺度建模

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