发表机构
University of Electronic Science and Technology of China（电子科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对PINN在陡梯度等问题上的谱偏差与约束冲突，提出GAC-PINN框架，结合自适应网格映射、硬约束拟设、傅里叶特征与算子感知路由，在多个基准上显著提升精度。
AI 中文摘要
对于具有陡峭梯度、尖锐界面或严重时空耦合的系统，物理信息神经网络（PINN）存在谱偏差、几何不灵活性和边界约束冲突等问题，这些问题损害了其精度和收敛性。为克服这些难题，我们提出了一种几何自适应与约束增强的物理信息神经网络（GAC-PINN）。该框架由四个组件构成：用于微分同胚点集中并带有雅可比正则化的梯度驱动自适应网格映射（AGM）；具有空间变化边界过渡宽度的自适应带宽硬约束拟设；作为谱预处理器以进一步增强高波数表示的Gaussian傅里叶特征映射；以及一个算子感知路由器，该路由器根据控制偏微分方程是否包含时间导数自动选择合适的硬约束构造方式。AGM回调机制和三阶段训练策略确保了各组件间的稳定协调。包括粘性Burgers方程、尖峰二维泊松问题和Allen-Cahn相变方程在内的基准测试表明，GAC-PINN分别达到了(1.747±0.450)×10⁻⁴、(2.868±0.947)×10⁻⁵和(1.756±0.712)×10⁻³的相对L²误差，始终优于基线方法。消融研究进一步揭示，仅使用AGM即可获得比基于残差的自适应细化（RAR）显著更低的误差，而RAR仅在结合FFM时才变得有益，这展示了上下文相关的模块交互。收敛性分析验证了随着分辨率提高，误差快速降低并趋于饱和，从而为应用力学和计算物理中具有局部尖锐特征问题的高保真模拟建立了一个实用的自适应框架。
英文摘要
For systems with steep gradients, sharp interfaces, or severe spatio-temporal coupling, Physics-informed neural networks (PINNs) suffer from spectral bias, geometric inflexibility, and boundary constraint conflicts, which undermine accuracy and convergence. To overcome these issues, we propose a geometry-adaptive and constraint-enhanced PINN (GAC-PINN). The framework comprises four components: a gradient-driven adaptive grid mapping (AGM) for diffeomorphic point concentration with Jacobian regularization, an adaptive bandwidth hard-constraint ansatz with spatially-varying boundary transition widths, a Gaussian Fourier feature mapping as a spectral preconditioner to further enhance high-wavenumber representation, and an operator-aware router that automatically selects the appropriate hard-constraint construction based on whether the governing PDE contains temporal derivatives. An AGM callback mechanism and a three-stage training strategy ensure stable coordination. Benchmarks including the viscous Burgers equation, a sharp-peaked 2D Poisson problem, and the Allen-Cahn phase-transition equation show that GAC-PINN attains relative (L^2) errors of ((1.747\pm 0.450)\times 10^{-4}), ((2.868\pm 0.947)\times 10^{-5}), and ((1.756 \pm 0.712)\times 10^{-3}), respectively, consistently outperforming the baselines. Ablation studies further reveal that AGM alone yields a substantially lower error than residual-based adaptive refinement (RAR), while RAR becomes beneficial only when combined with FFM, demonstrating a context-dependent module interaction. Convergence analysis verifies rapid error reduction and saturation with increasing resolution, establishing a practical adaptive framework for high-fidelity simulation of problems with localized sharp features in applied mechanics and computational physics.
Comments21 pages, 9 figures, 5 Tables
