On the extreme eigenvalues of the Gram Matrix in Physics-Informed Neural Networks for the Poisson Equation

Abstract

The smallest and largest eigenvalues of the Gram matrix induced by the differential neural tangent kernel (DNTK) play a pivotal role in the analysis of over-parameterized PINNs trained by gradient type algorithms. However, a theoretical analysis of the extreme eigenvalues remains completely absent due to the challenge posed by the presence of multiple differential operators. In this work, we provide explicit lower and upper bounds for the extreme eigenvalues of the infinite DNTK matrix for the Poisson equation with the Dirichlet boundary condition for two-layer RePU neural networks without the bias term. The setting is fairly general with respect to the sampling points and input dimension \(d\): \(δ\)-separated and additionally \(d\geq 3\) when deriving the lower bound of the smallest eigenvalue. These results extend that for the neural tangent kernel, and to the best of our knowledge, represent the first results on the spectrum of the DNTK.