AI 中文总结
本研究提出双分支频谱门控架构DBSG-PINN，通过消融实验发现频率分解仅在频谱复杂的物理信息神经网络基准问题上可显著降低误差，在平滑问题上益处有限。
AI 中文摘要
偏微分方程（PDE）常具有神经网络难以近似的高频与多尺度特征。物理信息神经网络（PINNs）将控制方程直接嵌入训练过程，但存在频谱偏差：其学习低频分量的速度快于高频分量。傅里叶特征嵌入、正弦激活等技术可解决该问题，但多数研究假设这些技术普遍有效，未验证其在哪些频谱区间真正发挥作用。我们提出一种双分支频谱门控架构（DBSG-PINN），将低频与高频分量拆分至独立子网络，由自适应门连接，并用该架构对频率分解与频谱路由开展部分受控消融实验。我们在5个一维基准PDE上测试该架构，涵盖从平滑单尺度问题到振荡多尺度问题。频率分解在频谱复杂的基准问题上效果最佳，在多模态波动问题上可将相对L₂误差降低多达59.2%，但在更平滑的PDE上几乎无益处。在其中一个基准问题（一维波动）上，其性能显著逊于更简单的固定组合变体。门控的益处随目标解的频谱丰富度提升而增大：全模型相比消融模型的优势在多尺度基准问题上最大，在单尺度基准问题上最小（甚至为负），这与门控利用频率结构而非充当噪声的结论一致，不过本研究未直接可视化或量化其空间激活。所有结果均来自5个一维基准问题的单一训练种子，因此本研究为探索性研究，旨在提出问题而非解答问题，并概述了验证该模式是否成立所需的额外种子与基准问题。
英文摘要
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.
