Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?
This work addresses theoretical convergence issues in PINNs for PDEs, which is incremental as it builds on existing NTK analysis.
The paper tackled the convergence of the neural tangent kernel (NTK) in physics-informed neural networks (PINNs) for general partial differential equations (PDEs), finding that the homogeneity of differential operators is crucial for convergence, validated through examples like the sine-Gordon and KdV equations.
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.