Residual-Based Adaptive Refinement in Hybrid Variational-Collocation PINNs for Singular Elliptic PDEs with Mixed Dirichlet-Neumann Boundary Conditions
Solutions of second-order elliptic PDE problems on nonconvex or cracked domains with mixed Dirichlet-Neumann conditions typically lie outside $H^2(Ω)$. Near re-entrant corners, crack tips and points where the boundary condition changes type, the gradient grows like $r^{λ-1}$ with $λ<1$. We study residual-based adaptive refinement in hybrid variational-collocation PINNs. A multiscale Fourier-feature network is trained on a composite loss. The loss combines a variational residual tested against Legendre bubble functions, a strong-form collocation term and penalties for the Dirichlet and Neumann data. The variational term needs only first derivatives of the network, whereas the collocation t...
| Authors | Somnath Giri, Muslim Malik, Rishikesh Yadav |
|---|---|
| Primary category | math.NA |
| Published | 19 October 2026 |
| Last updated at source | 19 October 2026 |
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