Adaptive refinement improves solving hard elliptic pdes near corners and cracks
Residual-Based Adaptive Refinement in Hybrid Variational-Collocation PINNs for Singular Elliptic PDEs with Mixed Dirichlet-Neumann Boundary Conditions
Machine Learning
Summary
Some mathematical problems called elliptic PDEs describe how things like heat or stress spread, but their solutions get tricky near sharp corners, cracks, or places where boundary rules change. The authors studied a smart approach that combines two ways to check the solution's accuracy and selectively focuses effort where the problem is hardest. Their method uses special neural networks and picks new points to improve accuracy based on how big the mistake is. Tests on tough 2D problems showed this adaptive method can reduce errors dramatically compared to spreading effort evenly.
What this means in practice
- •For computational engineers: Enhance simulation accuracy for fracture mechanics and material stress analysis by focusing computations near singularities and boundary transitions.
- •For numerical software developers: Incorporate adaptive point refinement techniques in PDE solvers to reduce computational cost while improving solution precision in difficult geometry regions.
Authors
Somnath Giri, Muslim Malik, Rishikesh Yadav
Abstract
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 term requires second derivatives and provides a residual that can be evaluated at any point. After each training stage, candidate points selected by this residual are added to the collocation set. The method is tested on five two dimensional benchmarks, namely a Mode~III crack, a cross-shaped domain with four re-entrant corners with unequal amplitude, a reaction-diffusion problem with four localized peaks, a monotone semilinear problem with a boundary layer and the Motz problem. Compared with uniform point insertion under the same collocation and iteration budget, residual-based refinement reduced the relative $L^2$ error by up to a factor of $32$, the relative $L^\infty$ error by up to a factor of $38$ and the relative $H^1$ semi-norm error by up to a factor of $15$.