Quasi-Optimal Least Squares: Inhomogeneous boundary conditions, and application with machine learning

TL;DR

This paper develops a least squares approach for PDEs with inhomogeneous boundary conditions, ensuring quasi-best approximations and stable finite element and machine learning solutions using adversarial networks.

Contribution

It introduces a novel least squares formulation that handles inhomogeneous boundary conditions without fractional Sobolev norms, applicable to finite elements and machine learning.

Findings

Constructed stable finite element pairs for PDEs.

Applied adversarial networks for machine learning solutions.

Achieved quasi-best approximation properties.

Abstract

We construct least squares formulations of PDEs with inhomogeneous essential boundary conditions, where boundary residuals are not measured in unpractical fractional Sobolev norms, but which formulations nevertheless are shown to yield a quasi-best approximations from the employed trial spaces. Dual norms do enter the least-squares functional, so that solving the least squares problem amounts to solving a saddle point or minimax problem. For finite element applications we construct uniformly stable finite element pairs, whereas for Machine Learning applications we employ adversarial networks.