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

• URL: http://arxiv.org/abs/2606.02475v2
• Date: Fri, 05 Jun 2026 09:36:44 GMT
• Title: Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers
• Abstract summary: 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.<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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-Dörfler 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 Schrödinger 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.

Related papers

• Neural-network solution of subtracted three-body Faddeev integral equations near the Efimov limit [0.0]
  Deep-neural-network (DNN) ansatz to the symmetrized spectator vector of the subtracted three-body Faddeev integral equation for identical bosons near the Efimov limit.<n>Network is trained by minimizing the residual of the discretized integral equation.
  arXiv  Detail & Related papers  (2026-06-09T02:52:41Z)
• Fourier Feature Pyramids for Physics-Informed Neural Networks [47.75068288153803]
  We present an improved neural field architecture for solving partial differential equations (PDEs)<n>We call our model beignet (Bandlimited Embedding with Interpolated Grid Network)<n>We demonstrate that beignet finds significantly more accurate solutions on PDE benchmarks using fewer parameters than state-of-the-art PINN methods.
  arXiv  Detail & Related papers  (2026-05-22T23:12:56Z)
• Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations [0.0]
  HIN-LRI is a hybrid framework that augments a classical numerical solver with a neural operator trained to correct the solver's structured truncation error.<n>A lightweight neural network, operating on a low-dimensional latent manifold, learns the residual defect that analytical methods cannot close.<n>Experiments show that HIN-LRI improves accuracy over analytical, splitting methods, and neural PDE surrogates, with stable spatial refinement, effective out-of-distribution transfer, and modest online overhead.
  arXiv  Detail & Related papers  (2026-05-06T12:50:36Z)
• Auxiliary Finite-Difference Residual-Gradient Regularization for PINNs [0.0]
  We study a hybrid design in which the governing PDE residual remains automatic-differentiation based, while finite differences appear only in a weak auxiliary term.<n>The FD term regularizes the residual field without replacing the PDE residual itself.<n>Across seeds 0-5 and 100k epochs, the most reliable tested configuration is a fixed shell weight of 5e-4 under the Kourkoutas-beta regime.
  arXiv  Detail & Related papers  (2026-04-15T23:09:19Z)
• Neural PDE Solvers with Physics Constraints: A Comparative Study of PINNs, DRM, and WANs [1.131316248570352]
  Partial equations (PDEs) underpin models across science and engineering, yet analytical solutions are atypical and classical mesh-based solvers can be costly in high dimensions.<n>This dissertation presents a unified comparison of three mesh-free neural PDE solvers, physics-informed neural networks (PINNs), the deep Ritz method (DRM), and weak adversarial networks (WANs), on Poisson problems (up to 5D) and the time-independent Schr"odinger equation in 1D/2D.
  arXiv  Detail & Related papers  (2025-10-09T13:41:51Z)
• Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation [0.0]
  This paper systematically benchmarks RBF-PIELM, a rapid PINN variant-an extreme learning machine with radial-basis activations-for higher-order PDEs.<n>Our results show up to $(350times)$ faster training than PINNs and over $(10times)$ fewer parameters for comparable solution accuracy.
  arXiv  Detail & Related papers  (2025-10-06T04:54:04Z)
• Decentralized Nonconvex Composite Federated Learning with Gradient Tracking and Momentum [78.27945336558987]
  Decentralized server (DFL) eliminates reliance on client-client architecture.<n>Non-smooth regularization is often incorporated into machine learning tasks.<n>We propose a novel novel DNCFL algorithm to solve these problems.
  arXiv  Detail & Related papers  (2025-04-17T08:32:25Z)
• Enabling Automatic Differentiation with Mollified Graph Neural Operators [73.52999622724101]
  We propose the mollified graph neural operator ($m$GNO), the first method to leverage automatic differentiation and compute exact gradients on arbitrary geometries.<n>For a PDE example on regular grids, $m$GNO paired with autograd reduced the L2 relative data error by 20x compared to finite differences.<n>It can also solve PDEs on unstructured point clouds seamlessly, using physics losses only, at resolutions vastly lower than those needed for finite differences to be accurate enough.
  arXiv  Detail & Related papers  (2025-04-11T06:16:30Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Gradient-Free Methods for Deterministic and Stochastic Nonsmooth Nonconvex Optimization [94.19177623349947]
  Non-smooth non optimization problems emerge in machine learning and business making. Two core challenges impede the development of efficient methods with finitetime convergence guarantee. Two-phase versions of GFM and SGFM are also proposed and proven to achieve improved large-deviation results.
  arXiv  Detail & Related papers  (2022-09-12T06:53:24Z)
• Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions [68.8204255655161]
  We develop a physics-informed neural network (PINN) model for parabolic problems with a sharply perturbed initial condition. Localized large gradients in the ADE solution make the (common in PINN) Latin hypercube sampling of the equation's residual highly inefficient. We propose criteria for weights in the loss function that produce a more accurate PINN solution than those obtained with the weights selected via other methods.
  arXiv  Detail & Related papers  (2022-08-18T05:00:24Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.