做偏微分方程数值模拟的团队，可以用PINN残差替代传统误差估计器来指导网格自适应，显著节省计算资源——60个自由度就能达到192个自由度的精度，值得在工程仿真中试试。
该研究提出一种混合策略，利用物理信息神经网络（PINN）作为离网残差探针，为有限差分求解器提供自适应网格细化（AMR）指导。PINN在域内采样残差并转换为单元级指示器，引导网格加密，最终由经典有限差分求解器完成近似计算。在一维粘性Burgers方程测试中，PINN阈值细化方法仅用60个自由度即达到0.021067的相对L²误差，而均匀细化需192个自由度才达到0.022617，误差降低约67.5%。在2D和3D代理测试中，PINN残差能组织结构化细化并优于随机细化，但未持续超越梯度指示器。该方法将物理信息诊断能力融入经典求解器，在保持可靠性的同时提升计算效率。
Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers
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.