PINNs and GaLS: An Priori Error Estimates for Shallow Physically Informed Neural Network Applied to Elliptic Problems

Recently Physically Informed Neural Networks have gained more and more popularity to solve partial differential equations, given the fact they escape the course of dimensionality. First Physically Informed Neural Networks are viewed as an underdetermined point matching collocation method then we expose the connection between Galerkin Least Square (GALS) and PINNs, to develop an a priori error estimate, in the context of elliptic problems. In particular techniques that belong to the realm of the least square finite elements and Rademacher complexity analysis will be used to obtain the above mentioned error estimate.