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

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

Problem Statement
Classical finite-difference solvers are reliable for solving partial differential equations (PDEs), but their efficiency is often compromised by the need for uniform mesh refinement. When solutions exhibit localized features such as sharp gradients, fronts, or oscillations, uniform refinement wastes degrees of freedom (DOFs) in regions where high resolution is unnecessary. While Adaptive Mesh Refinement (AMR) addresses this by concentrating resolution where errors are high, traditional error indicators rely on mesh-dependent quantities (e.g., gradients, jumps, or recovery-based estimates). This paper investigates whether Physics-Informed Neural Network (PINN) residuals, which can be evaluated off-grid via automatic differentiation, can serve as effective indicators to guide mesh adaptation for classical solvers, without replacing the solvers themselves.

Methodology
The proposed approach treats the PINN not as a standalone solver, but as a "residual probe" to inform a classical finite-difference solver. The workflow follows a standard solve–estimate–mark–refine loop:

1. PINN Training: A neural network $u_\theta$ is trained (or partially trained) to approximate the solution by minimizing a composite loss function comprising the PDE residual, boundary conditions, and initial conditions.
2. Residual Extraction: The pointwise PINN residual $r_\theta(x,t) = \mathcal{N}[u_\theta] - f$ is computed at arbitrary points within the domain using automatic differentiation.
3. Cellwise Indicator Construction: For each cell $K$ in the current mesh, a refinement indicator $\eta_K$ is computed by aggregating the squared residual magnitude over a set of probe points within the cell, weighted by quadrature weights and scaled by the cell measure $|K|$.
4. Marking and Refinement: Cells are marked for refinement using either:
   • Threshold Marking: Selecting cells where $\eta_K \ge \tau \eta_{\max}$.
   • Dörfler Marking: Selecting a minimal set of cells carrying a prescribed fraction $\vartheta$ of the total indicator mass.
5. Classical Solve: The marked cells are refined (e.g., via bisection or subdivision), and the final PDE solution is computed using a classical non-uniform finite-difference solver on the adapted mesh.

Key Contributions
The paper makes three primary contributions:

1. Formulation of a Hybrid Workflow: It establishes a specific pipeline where a PINN residual field is converted into cellwise refinement indicators for classical solvers, separating the diagnostic role of the neural network from the numerical approximation role of the finite-difference method.
2. Bounded Claims on Utility: It explicitly distinguishes between using a PINN residual as a refinement signal versus claiming it as a certified global error bound. The method is framed as a diagnostic tool to transfer physics-informed information into mesh adaptation, preserving the classical solver as the final engine.
3. Comprehensive Benchmarking: The strategy is evaluated against uniform refinement, random refinement, gradient-based indicators, reference-guided diagnostics, and standalone PINN approximations across three distinct benchmarks.

Experimental Results
The method was tested on three benchmarks with varying dimensions and validation modes:

• 1D Viscous Burgers Equation (Full-Solver Validation):

  • This was the primary validation where the adapted mesh was passed to a non-uniform finite-difference solver.
  • Performance: PINN-threshold refinement achieved a final relative $L_2$ error of 0.021067 using 60 DOFs. In contrast, uniform refinement required 192 DOFs to reach an error of 0.022617.
  • Comparison: At matched DOFs (60), uniform refinement yielded an error of 0.064813, demonstrating a significant improvement (~67.5% reduction) by the PINN-guided method.
  • Limitation: The gradient-based indicator slightly outperformed the PINN methods (error 0.019435 at 57 DOFs), indicating the PINN approach is competitive but not universally superior to classical gradient indicators in this setting.

• 2D Nonlinear Schrödinger Equation (Manufactured Proxy Test):

  • Used to test residual localization for coupled oscillatory fields.
  • Performance: PINN-Dörfler refinement significantly outperformed random refinement and standalone PINN approximation. However, it did not outperform the gradient-based or uniform refinement baselines in terms of proxy error.

• 3D Incompressible Navier–Stokes System (Manufactured Proxy Test):

  • Used to test residual localization in a coupled velocity–pressure system.
  • Performance: PINN-guided variants improved over uniform and random refinement. The gradient indicator remained the most accurate in absolute proxy error, though it required more DOFs than the PINN-guided methods.

Significance and Claims
The paper concludes with a modest, bounded interpretation of its findings:

• Utility: PINN-guided AMR is a useful hybrid strategy, particularly when uniform refinement is inefficient or when a trained PINN residual field is already available. It successfully transfers off-grid physics information to guide classical mesh adaptation.
• Limitations: The method is not presented as a universal replacement for established adaptive error estimators (such as gradient, jump, or adjoint-based indicators). In the 1D Burgers test, the gradient indicator remained superior in absolute accuracy. In higher-dimensional proxy tests, the PINN residual was informative but not consistently the best indicator.
• Role: The PINN serves as an auxiliary diagnostic signal. The classical solver retains responsibility for the final discretized approximation, stability, and convergence.
• Scope: The strongest evidence is limited to the 1D Burgers full-solver validation. The 2D and 3D results are "manufactured proxy tests" that demonstrate residual localization capabilities but do not claim superiority in full adaptive finite-difference or finite-volume solver performance for complex engineering problems.

The paper argues that the value of this approach lies in its ability to leverage neural networks for diagnostic purposes (identifying under-resolved regions) while relying on mature, stable classical methods for the numerical solution, thereby avoiding the training instabilities and reliability issues often associated with using PINNs as standalone solvers.