研究PINNs的神经正切核在求解偏微分方程时是否收敛，发现微分算子齐次性是关键。
Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?
- 通过分析神经正切核的初始化与训练过程，揭示其收敛机制。
- 证明微分算子的齐次性对神经正切核收敛起决定性作用。
- 在正弦-戈登和KdV方程上验证了理论结论的有效性。
本文研究基于物理信息神经网络（PINNs）的通用偏微分方程（PDEs）的神经正切核（NTK）。众所周知，人工神经网络的训练可转化为NTK的演化过程。我们分析了NTK的初始化及其在训练过程中对一般PDE的收敛条件。理论结果表明，微分算子的齐次性在NTK收敛中起关键作用。此外，基于PINNs，我们通过正弦-戈登方程的初值问题和KdV方程的初边值问题验证了NTK的收敛条件。
原文摘要 · Abstract (English)
In this paper, we study the neural tangent kernel (NTK) for general partial differential equations (PDEs) based on physics-informed neural networks (PINNs). As we all know, the training of an artificial neural network can be converted to the evolution of NTK. We analyze the initialization of NTK and the convergence conditions of NTK during training for general PDEs. The theoretical results show that the homogeneity of differential operators plays a crucial role for the convergence of NTK. Moreover, based on the PINNs, we validate the convergence conditions of NTK using the initial value problems of the sine-Gordon equation and the initial-boundary value problem of the KdV equation.
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