🤖 AI Summary
This work addresses the inefficiency of traditional finite difference methods on uniform grids in resolving localized complex features—such as shocks or steep gradients—without excessive degrees of freedom. The authors propose a hybrid adaptive strategy that leverages the residual of a physics-informed neural network (PINN) as an off-grid indicator to generate cell-wise refinement metrics. These metrics drive non-uniform mesh refinement through a combination of Dörfler and threshold-based marking criteria, while high-fidelity solutions are still computed using finite differences. Innovatively, the PINN serves not as a solver but as an adaptive guidance tool, effectively integrating physical priors with classical numerical methods. In one-dimensional viscous Burgers’ equation, the approach achieves superior accuracy with only 60 degrees of freedom—outperforming a uniform 192-point grid by reducing error by 67.5%—and demonstrates marked advantages over random refinement strategies in two- and three-dimensional proxy problems.
📝 Abstract
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.