An operator preconditioning perspective on training in physics-informed machine learning

arXiv:2310.05801v2 Announce Type: replace
Abstract: In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in …

abstract algorithms arxiv behavior cs.lg differential gradient key machine machine learning paper perspective physics physics-informed training type