Solving partial differential equations (PDEs) by neural networks as well as Kolmogorov-Arnold Networks (KANs), including physics-informed neural networks (PINNs), physics-informed KANs (PIKANs), and neural operators, are known to exhibit spectral bias, whereby low-frequency components of the solution are learned significantly faster than high-frequency modes. While spectral bias is often treated as an intrinsic representational limitation of neural architectures, its interaction with optimization dynamics and physics-based loss formulations remains poorly understood. In this work, we provide a systematic investigation of spectral bias in physics-informed and operator learning frameworks, with emphasis on the coupled roles of network architecture, activation functions, loss design, and optimization strategy. We quantify spectral bias through frequency-resolved error metrics, Barron-norm diagnostics, and higher-order statistical moments, enabling a unified analysis across elliptic, hyperbolic, and dispersive PDEs. Through diverse benchmark problems, including the Korteweg-de Vries, wave and steady-state diffusion-reaction equations, turbulent flow reconstruction, and earthquake dynamics, we demonstrate that spectral bias is not simply representational but fundamentally dynamical. In particular, second-order optimization methods substantially alter the spectral learning order, enabling earlier and more accurate recovery of high-frequency modes for all PDE types. For neural operators, we further show that spectral bias is dependent on the neural operator architecture and can also be effectively mitigated through spectral-aware loss formulations without increasing the inference cost.

翻译：利用神经网络及Kolmogorov-Arnold网络（KANs）求解偏微分方程（PDEs）的方法，包括物理信息神经网络（PINNs）、物理信息KANs（PIKANs）以及神经算子，已知存在谱偏置现象，即解的低频分量被学习的速度显著快于高频模态。虽然谱偏置常被视为神经架构固有的表示能力限制，但其与优化动力学及基于物理的损失函数设计之间的相互作用仍不甚明晰。本研究对物理信息与算子学习框架中的谱偏置进行了系统性探究，重点关注网络架构、激活函数、损失函数设计及优化策略的耦合作用。我们通过频率分辨误差度量、Barron范数诊断以及高阶统计矩对谱偏置进行量化，实现了对椭圆型、双曲型及色散型PDE的统一分析。通过包括Korteweg-de Vries方程、波动方程与稳态扩散-反应方程、湍流重建以及地震动力学在内的多样化基准问题，我们证明谱偏置不仅是表示层面的问题，更是动力学层面的根本性问题。特别地，二阶优化方法显著改变了谱学习顺序，使得所有类型PDE的高频模态能够更早且更精确地被恢复。对于神经算子，我们进一步表明谱偏置依赖于神经算子的架构，并且可以通过谱感知的损失函数设计有效缓解，而无需增加推理成本。