Spectral-Bias and Kernel-Task Alignment in Physically Informed Neural Networks
Abstract
Physically informed neural networks (PINNs) are a promising emerging method for solving differential equations. As in many other deep learning approaches, the choice of PINN design and training protocol requires careful craftsmanship. Here, we suggest a comprehensive theoretical framework that sheds light on this important problem. Leveraging an equivalence between infinitely over-parameterized neural networks and Gaussian process regression (GPR), we derive an integro-differential equation that governs PINN prediction in the large data-set limit -- the neurally-informed equation. This equation augments the original one by a kernel term reflecting architecture choices and allows quantifying implicit bias induced by the network via a spectral decomposition of the source term in the original differential equation.
Cite
@article{arxiv.2307.06362,
title = {Spectral-Bias and Kernel-Task Alignment in Physically Informed Neural Networks},
author = {Inbar Seroussi and Asaf Miron and Zohar Ringel},
journal= {arXiv preprint arXiv:2307.06362},
year = {2023}
}