算子感知的傅里叶特征物理信息神经网络初始化

中文摘要 AI 辅助

物理信息神经网络（PINNs）通常表现出频谱偏差，即目标函数的某些频率收敛速度比其他频率慢。在本工作中，我们在神经正切核（NTK）框架下分析傅里叶特征PINNs的训练动态，以解决这一局限性。我们推导出一个显式的演化方程来估计频域中的残差误差，表明特定频率的收敛速率主要由微分算子符号与初始化权重的谱密度的乘积决定。利用这一理论洞见，我们提出了一种信息丰富的初始化策略，该策略针对所求解的具体偏微分方程（PDE）定制初始权重分布。通过这种方法，我们可以减小算子引起的频谱偏差，平衡整个频谱上的收敛速率，并获得更好的预测精度。在线性和非线性偏微分方程上的数值实验证实，与标准初始化方法相比，这种初始化策略改善了跨频率的学习动态和逼近精度，且无需额外的训练成本。

英文摘要

Physics-Informed Neural Networks (PINNs) typically exhibit spectral bias, where some frequencies of the target function converge more slowly than others. In this work, we analyze the training dynamics of Fourier Feature PINNs in the Neural Tangent Kernel regime to address this limitation. We derive an explicit evolution equation to estimate the residual error in the frequency domain, demonstrating that the convergence rate of specific frequencies is primarily governed by the product of the differential operator's symbol and the spectral density of the initialization weights. Leveraging this theoretical insight, we propose an informative initialization strategy that tailors the initial weight distribution to the specific PDE being solved. With this method, we can diminish the operator-induced spectral bias, balancing the convergence rates across the frequency spectrum and achieving better prediction accuracy. Numerical experiments on linear and nonlinear partial differential equations confirm that this initialization strategy improves learning dynamics and approximation accuracy across frequencies compared to standard initialization methods, with no additional training cost.

发表机构

• National Center for Artificial Intelligence（国家人工智能中心）
• University of Pennsylvania（宾夕法尼亚大学）
• Pontificia Universidad Católica de Chile（智利天主教大学）
• Millennium Institute for Intelligent Healthcare Engineering, iHEALTH（千禧智能医疗工程研究所）

机构由 AI 辅助整理，请以论文原文为准。