A Neural-Enhanced Weak Galerkin Method for Second-Order Elliptic Problems with Low-Regularity Solutions

TL;DR

This paper introduces a neural-enhanced weak Galerkin method for second-order elliptic problems, improving approximation of low-regularity solutions while maintaining stability and convergence properties.

Contribution

It combines neural network functions with weak Galerkin finite elements, preserving variational structure and achieving better accuracy for singular solutions.

Findings

Establishes a quasi-optimal error estimate in a discrete WG energy norm.

Retains optimal convergence rates for smooth solutions.

Effectively captures singular components, improving accuracy over standard WG methods.

Abstract

We propose a neural-enhanced weak Galerkin (WG) finite element method for second-order elliptic problems with low-regularity solutions. The method augments the classical WG approximation space with neural network functions constructed via a residual-driven Galerkin enrichment procedure. This approach preserves the variational structure, symmetry, and stability of the WG formulation while enhancing its ability to approximate non-smooth and singular solution components. We establish a quasi-optimal error estimate in a discrete WG energy norm, incorporating both projection and consistency errors. In particular, the method retains optimal convergence rates for smooth solutions. For problems admitting a regular--singular decomposition, we further show that the neural enrichment effectively captures the singular component, yielding improved accuracy over standard WG methods.