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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2604.06115 |
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| _version_ | 1866910109440933888 |
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| author | Wang, Chunmei |
| author_facet | Wang, Chunmei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06115 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Neural-Enhanced Weak Galerkin Method for Second-Order Elliptic Problems with Low-Regularity Solutions Wang, Chunmei Numerical Analysis 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. |
| title | A Neural-Enhanced Weak Galerkin Method for Second-Order Elliptic Problems with Low-Regularity Solutions |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2604.06115 |