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arXiv · 2604.06115

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

Chunmei Wang
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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.

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Chunmei Wang. 2026-04-07. A Neural-Enhanced Weak Galerkin Method for Second-Order Elliptic Problems with Low-Regularity Solutions. https://arxiv.org/abs/2604.06115

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