Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers
2026-06-01 • Computational Engineering, Finance, and Science
Computational Engineering, Finance, and ScienceMachine Learning
AI summaryⓘ
The authors explore a new way to improve how computer programs solve complex equations by focusing more detail only where it's really needed. Instead of using neural networks as the main solver, they use them to spot tricky parts in the problem and guide where to add more detail in the computation grid. This method was tested on different problems and showed it can achieve similar or better accuracy with fewer computing points than traditional methods that refine the grid evenly. While sometimes other methods perform slightly better, their approach effectively uses neural network insights to help traditional solvers be more efficient.
finite-difference solverpartial differential equationsadaptive mesh refinementphysics-informed neural network (PINN)residual indicatorBurgers equationnonlinear Schrödinger equationNavier–Stokes equationserror analysis
Authors
Henry Kasumba, Ronald Katende
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.