发现单层物理信息网络在非线性PDE求解中的宽度失效现象。
Scaling Laws and Pathologies of Single-Layer PINNs: Network Width and PDE Nonlinearity
- 通过实验揭示单层PINN的优化失败，宽度增加无法降低误差。
- 非线性越强，误差下降越困难，存在双重优化瓶颈。
- 提出测量复杂缩放效应的方法，适合研究神经微分方程者。
我们在典型非线性偏微分方程上建立了单层物理信息神经网络的实证缩放规律。识别出两种优化失败：(i) 基线病态，即在固定非线性下，网络宽度增加时解误差不降反增，低于理论逼近界；(ii) 病态加剧，非线性越强，该失败越严重。我们提供定量证据表明，简单的可分离幂律不足以描述，缩放行为由更复杂的非可分离关系主导。该现象与谱偏差一致，即网络难以学习随非线性增强的高频解成分。我们证明优化而非近似能力是主要瓶颈，并提出一种方法来实证测量这些复杂缩放效应。
原文摘要 · Abstract (English)
We establish empirical scaling laws for Single-Layer Physics-Informed Neural Networks on canonical nonlinear PDEs. We identify a dual optimization failure: (i) a baseline pathology, where the solution error fails to decrease with network width, even at fixed nonlinearity, falling short of theoretical approximation bounds, and (ii) a compounding pathology, where this failure is exacerbated by nonlinearity. We provide quantitative evidence that a simple separable power law is insufficient, and that the scaling behavior is governed by a more complex, non-separable relationship. This failure is consistent with the concept of spectral bias, where networks struggle to learn the high-frequency solution components that intensify with nonlinearity. We show that optimization, not approximation capacity, is the primary bottleneck, and propose a methodology to empirically measure these complex scaling effects.
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。