Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers
Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural network (PINN) is used not as the final solver, but as an off-grid residual probe for adaptive mesh refinement. The PINN residual is sampled over the domain, converted into cellwise indicators, and used to guide refinement before the final approximation is computed by a finite-difference solver. The me...
| Authors | Henry Kasumba, Ronald Katende |
|---|---|
| Primary category | math.NA |
| Published | 1 June 2026 |
| Last updated at source | 5 June 2026 |
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