Extended Galerkin neural network approximation of singular variational problems with error control

TL;DR

This paper introduces extended Galerkin neural networks (xGNN), a variational approach with error control for approximating boundary value problems, especially those with singular solutions, demonstrated on fluid flow problems.

Contribution

The work develops a rigorous variational framework and an extended neural network architecture capable of learning singular solution structures, enhancing approximation accuracy.

Findings

Effective approximation of singular solutions in fluid flow problems.

Numerical results demonstrate improved accuracy and robustness.

Framework provides error control in neural network approximation.

Abstract

We present extended Galerkin neural networks (xGNN), a variational framework for approximating general boundary value problems (BVPs) with error control. The main contributions of this work are (1) a rigorous theory guiding the construction of new weighted least squares variational formulations suitable for use in neural network approximation of general BVPs (2) an ``extended'' feedforward network architecture which incorporates and is even capable of learning singular solution structures, thus greatly improving approximability of singular solutions. Numerical results are presented for several problems including steady Stokes flow around re-entrant corners and in convex corners with Moffatt eddies in order to demonstrate efficacy of the method.