A Neural-Enhanced Weak Galerkin Method for Second-Order Elliptic Problems with Low-Regularity Solutions
This is an incremental improvement for computational mathematics, specifically for solving partial differential equations with non-smooth solutions.
The authors tackled the problem of solving second-order elliptic problems with low-regularity solutions by proposing a neural-enhanced weak Galerkin method that augments classical approximation spaces with neural network functions. The result shows that the method preserves optimal convergence rates for smooth solutions and effectively captures singular components, yielding improved accuracy over standard methods.
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.