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

Problem
Classical finite-difference solvers for partial differential equations (PDEs) often suffer from inefficiencies due to uniform mesh refinement, which can waste computational resources when solution difficulties are localized. This paper addresses the gap in adaptive mesh refinement (AMR) strategies by proposing a method that leverages physics-informed neural networks (PINNs) as residual probes to inform mesh adaptation. The authors highlight that existing literature lacks effective integration of PINNs in this context, particularly in enhancing the performance of traditional solvers without replacing them. This work is presented as a preprint and has not undergone peer review.

Method
The proposed method employs a hybrid strategy where a PINN is utilized to compute residuals over the computational domain. These residuals are then transformed into cellwise indicators that guide the adaptive refinement of the mesh before the final solution is computed using a finite-difference solver. The authors evaluate their approach on three benchmarks, with a primary focus on the one-dimensional viscous Burgers equation. The PINN is trained on the PDE’s governing equations, and the refinement process is based on a thresholding mechanism (PINN-threshold) and a Dörfler refinement strategy (PINN-Dörfler). The training compute and specific architecture details are not disclosed, but the method emphasizes the integration of physics-informed diagnostics into the AMR process.

Results
The results demonstrate that the PINN-threshold refinement achieves a final relative (L^2) error of (0.021067) using only (60) degrees of freedom, outperforming uniform refinement, which yields an error of (0.022617) with (192) degrees of freedom. At matched mesh sizes, the PINN-threshold method reduces the error by approximately (67.5\%). The PINN-Dörfler refinement also shows competitive performance with an error of (0.021264) using (58) degrees of freedom. While a gradient-based indicator remains slightly more accurate, the results indicate that PINN-guided AMR can effectively organize mesh refinement and improve upon random refinement strategies, although it does not consistently outperform gradient or uniform baselines.

Limitations
The authors acknowledge that while the PINN-guided AMR shows promise, it does not universally outperform traditional gradient-based methods or uniform refinement across all tested scenarios. Additionally, the reliance on the PINN’s training quality and the potential computational overhead associated with training and evaluating the neural network are not fully explored. The paper does not address the scalability of the method to higher-dimensional problems beyond the tested cases.

Why it matters
This work has significant implications for the field of computational physics and numerical analysis, particularly in enhancing the efficiency of finite-difference solvers through adaptive mesh refinement informed by machine learning techniques. The integration of PINNs as residual indicators represents a novel approach to transferring physics-informed insights into mesh adaptation processes, potentially leading to more efficient simulations in complex physical systems. This research contributes to the ongoing discourse on the intersection of deep learning and traditional numerical methods, as discussed in related works available on arXiv.