Numerical analysis of physics-informed neural networks and related
  models in physics-informed machine learning
Tim De Ryck Siddhartha Mishra
Physics-informed neural networks (PINNs) and their variants have been very
popular in recent years as algorithms for the numerical simulation of both
forward and inverse problems for partial differential equations. This article
aims to provide a comprehensive review of currently available results on the
numerical analysis of PINNs and related models that constitute the backbone of
physics-informed machine learning. We provide a unified framework in which
analysis of the various components of the error incurred by PINNs in
approximating PDEs can be effectively carried out. A detailed review of
available results on approximation, generalization and training errors and
their behavior with respect to the type of the PDE and the dimension of the
underlying domain is presented. In particular, the role of the regularity of
the solutions and their stability to perturbations in the error analysis is
elucidated. Numerical results are also presented to illustrate the theory. We
identify training errors as a key bottleneck which can adversely affect the
overall performance of various models in physics-informed machine learning.