PINNs and GaLS: A Priori Error Estimates for Shallow Physics Informed Neural Networks Applied to Elliptic Problems

TL;DR

This paper develops an a priori error estimate for shallow Physics Informed Neural Networks (PINNs) applied to elliptic PDEs by connecting them with Galerkin Least Squares methods and employing finite element and complexity analysis techniques.

Contribution

It introduces a novel theoretical framework linking PINNs with Galerkin Least Squares, providing rigorous error bounds for elliptic problems.

Findings

Established a connection between PINNs and GALS.

Derived an a priori error estimate for PINNs.

Applied finite element and Rademacher complexity techniques.

Abstract

Physics Informed Neural Networks (PINNs) have recently gained popularity for solving partial differential equations, given the fact they escape the curse of dimensionality. In this paper, we present Physics Informed Neural Networks as an underdetermined point matching collocation method then 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 least square finite elements and Rademacher complexity analysis are used to obtain the error estimate.