Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?

TL;DR

This paper investigates the convergence of neural tangent kernels in physics-informed neural networks for solving general PDEs, highlighting the importance of differential operator homogeneity and validating conditions through specific PDE examples.

Contribution

It provides theoretical analysis of NTK convergence conditions for PINNs applied to PDEs, emphasizing the role of operator homogeneity and validating findings with PDE case studies.

Findings

Homogeneity of differential operators is crucial for NTK convergence.

Convergence conditions are validated using sine-Gordon and KdV equations.

Theoretical analysis links NTK behavior to PDE properties.

Abstract

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.