发表机构
The Chinese University of Hong Kong（香港中文大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究针对物理信息神经网络（PINN），提出微分神经切线核（DNTK）理论框架，通过NTK分析PINN，建立了无限宽度DNTK的正性，为训练PINN的梯度型算法分析奠定基础。
AI 中文摘要
神经切线核（NTK）是分析过参数化神经网络训练动态的有力工具。最近，理论框架已扩展到用于求解线性偏微分方程的物理信息神经网络（PINN），这是一类非常流行的神经偏微分方程求解器。在分析中，相关NTK的正性起着基本作用。然而，由于存在多个微分算子，确定PINN的NTK的正性极具挑战性。在这项工作中，我们提出了一个新的理论框架，称为微分神经切线核（DNTK），通过NTK来分析PINN，并为包括RePU和光滑但非多项式激活函数在内的广泛一类激活函数，针对所有线性微分算子，建立了浅层和深层神经网络的无限宽度DNTK的正性。这些理论结果为训练PINN的梯度型算法分析奠定了基础。
英文摘要
The Neural Tangent Kernel (NTK) is one powerful tool for analyzing the training dynamics of neural networks in the over-parameterized regime. Recently, the theoretical framework has been extended to physics-informed neural networks (PINNs) for solving linear PDEs, one highly popular class of neural PDE solvers. In the analysis, the positivity of the associated NTK plays a fundamental role. However, establishing the positivity of the NTK for PINNs is highly challenging, due to the presence of multiple differential operators. In this work, we propose a new theoretical framework, called Differential Neural Tangent Kernel (DNTK), for analyzing PINNs through the lens of the NTK, and establish the positivity of the infinite width DNTK for both shallow and deep neural networks for a wide class of activation functions, including RePU and smooth but non-polynomial activations, for all linear differential operators. These theoretical results lay the foundation for the analysis of gradient type algorithms for training PINNs.
Comments31 pages, 1 figure
