Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers
For practitioners of finite-difference PDE solvers, this hybrid approach offers a way to leverage PINNs for mesh adaptation without replacing classical solvers, though gains are incremental and not universally superior to gradient-based indicators.
The paper proposes using a physics-informed neural network (PINN) as a residual probe to guide adaptive mesh refinement for finite-difference PDE solvers. On the 1D Burgers equation, PINN-threshold refinement achieves a relative L2 error of 0.021067 with 60 degrees of freedom, compared to 0.022617 with 192 degrees for uniform refinement, a 67.5% error reduction at matched mesh size.
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 method is evaluated on three benchmarks. The main full-solver validation uses the one-dimensional viscous Burgers equation with a nonuniform finite-difference solve on the adapted meshes. PINN-threshold refinement attains final relative $L^2$ error $0.021067$ with $60$ degrees of freedom, compared with $0.022617$ for uniform refinement with $192$ degrees of freedom. At matched mesh size, PINN-threshold reduces the error by about $67.5\%$. PINN-D"orfler refinement gives similar performance, with error $0.021264$ using $58$ degrees of freedom. A gradient indicator remains slightly more accurate, so the result supports usefulness rather than universal superiority. Manufactured 2D and 3D proxy tests, based on a nonlinear Schr"odinger equation and an incompressible Navier--Stokes system, show that PINN residuals can organise structured refinement and improve over random refinement, although they do not consistently outperform gradient or uniform baselines. The results support PINN-guided AMR as a residual-indicator strategy for transferring physics-informed diagnostic information into finite-difference mesh adaptation while preserving the classical solver as the final approximation engine.