Title: Leveraging Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers

While classical finite-difference solvers continue to serve as dependable instruments for solving partial differential equations, their computational efficiency hinges critically on the strategic placement of mesh resolution. Employing uniform refinement often results in a wastage of degrees of freedom, particularly when solution complexities are confined to specific areas such as sharp gradients, fronts, oscillations, or regions sensitive to constraints. This study investigates a hybrid approach wherein a physics-informed neural network (PINN) functions not as the ultimate solver, but rather as an off-grid residual probe to facilitate adaptive mesh refinement.

In this methodology, the PINN residual is sampled across the domain and transformed into cell-wise indicators. These indicators direct the refinement process prior to the final approximation being generated by a finite-difference solver. The efficacy of this technique was assessed using three benchmark tests. The primary validation of the full solver involved the one-dimensional viscous Burgers equation, utilizing a nonuniform finite-difference solution on the adapted meshes.

The results demonstrated that PINN-threshold refinement achieved a final relative $L^2$ error of $0.021067$ using only $60$ degrees of freedom. In contrast, uniform refinement required $192$ degrees of freedom to reach a higher error of $0.022617$. When compared at an equivalent mesh size, the PINN-threshold method reduced the error by approximately $67.5\%$. Similarly, PINN-D"orfler refinement exhibited comparable performance, yielding an error of $0.021264$ with $58$ degrees of freedom. Although a gradient indicator proved to be marginally more accurate, these findings affirm the utility of the approach, even if it does not guarantee universal superiority.

Additional proxy tests involving manufactured 2D and 3D scenarios—based on an incompressible Navier--Stokes system and a nonlinear Schr"odinger equation—indicated that PINN residuals are capable of organizing structured refinement and offering improvements over random refinement strategies. However, these tests showed that PINN methods do not consistently surpass gradient or uniform baselines. Ultimately, the outcomes endorse PINN-guided adaptive mesh refinement as a viable residual-indicator strategy, effectively transferring physics-informed diagnostic insights into finite-difference mesh adaptation while maintaining the classical solver as the core approximation engine.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC