# Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers

arXiv: https://arxiv.org/abs/2606.02475 · HTML: https://mlsanity.com/paper/2606.02475/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 35 references that are in the map.
- First posted on arXiv 1 Jun 2026.

## Builds on (most cited first, 10 of 10)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [h-analysis and data-parallel physics-informed neural networks](https://mlsanity.com/paper/2302.08835/) (2023)

## Abstract (arXiv)

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 L2 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.