关于基于小波的 Kolmogorov Arnold 网络中频率控制与谱偏置的研究：通往物理信息 KAN 之路
On the study of frequency control and spectral bias in Wavelet-Based Kolmogorov Arnold networks: A path to physics-informed KANs
AI总结：
本文研究了 Wav-KANs 中母小波频率与谱偏置的关系，通过分析神经正切核特征值增强网络高频收敛能力，有效缓解谱偏置，为解决振荡物理现象提供了新见解。
摘要：
谱偏置，即神经网络在训练初始阶段优先学习函数低频成分的倾向，在逼近具有高频细节的解时构成了重大挑战。该问题在广泛用于求解描述物理现象微分方程的物理信息神经网络（PINNs）中尤为显著。现有文献中，小波 Kolmogorov Arnold 网络（Wav-KANs）等成果已展现出在捕获低频和高频成分方面的良好效果。类似地，傅里叶特征（FF）也常被用于应对此挑战。然而，Wav-KANs 的理论基础，特别是母小波频率与谱偏置之间的关系，仍有待深入探索。对 Wav-KANs 如何处理高频项的更深入理解，可为解决抛物型、椭圆型和双曲型微分方程中遇到的振荡现象提供宝贵见解。在本研究中，我们分析了 Wav-KANs 的神经正切核（NTK）特征值，以增强其在高频成分上的收敛能力，从而有效缓解谱偏置。我们通过数值实验验证了理论发现，并在实验中探讨了标准 PINNs 和傅里叶特征等传统方法在处理多频问题时的局限性。
英文摘要：
Spectral bias, the tendency of neural networks to prioritize learning low-frequency components of functions during the initial training stages, poses a significant challenge when approximating solutions with high-frequency details. This issue is particularly pronounced in physics-informed neural networks (PINNs), widely used to solve differential equations that describe physical phenomena. In the literature, contributions such as Wavelet Kolmogorov Arnold Networks (Wav-KANs) have demonstrated promising results in capturing both low- and high-frequency components. Similarly, Fourier features (FF) are often employed to address this challenge. However, the theoretical foundations of Wav-KANs, particularly the relationship between the frequency of the mother wavelet and spectral bias, remain underexplored. A more in-depth understanding of how Wav-KANs manage high-frequency terms could offer valuable insights for addressing oscillatory phenomena encountered in parabolic, elliptic, and hyperbolic differential equations. In this work, we analyze the eigenvalues of the neural tangent kernel (NTK) of Wav-KANs to enhance their ability to converge on high-frequency components, effectively mitigating spectral bias. Our theoretical findings are validated through numerical experiments, where we also discuss the limitations of traditional approaches, such as standard PINNs and Fourier features, in addressing multi-frequency problems.
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