发表机构
Indian Institute of Technology, Patna; Worcester Polytechnic Institute（印度巴特那理工学院; 伍斯特理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出ENPINN，将梯度与变分结构融入PINN损失以提升尖锐梯度解的分辨率，经理论分析与多类数值实验验证其性能优于现有PINN变体。
AI 中文摘要
物理信息神经网络（PINNs）已成为求解偏微分方程（PDEs）的无网格替代传统数值方法，但它们捕捉尖锐梯度的能力有限，在求解边界层和内部层时会产生显著误差。本文提出能量范数增强型PINN（ENPINN），将梯度信息和变分结构融入损失函数，以提升层主导解的分辨率。首先分析两种相关形式：弱损失PINNs（WLPINNs），其将测试函数融入传统PINN残差；梯度增强PINNs（gPINNs），其在损失中加入PDE残差的空间导数。通过分析这些形式，明确它们在求解陡峭解梯度时的局限性，进而指导ENPINN的系统构建。从理论上建立能量范数误差与ENPINN损失的关系，表明适当修改的残差导数项对准确捕捉边界层至关重要；还证明存在能量误差任意小的神经网络近似，并推导相应导数界，为所提框架提供理论基础。通过与现有PINN变体的系统对比评估ENPINN性能，对比对象包括具有陡峭梯度的对流-扩散-反应问题，数值实验涵盖燃烧模型、耦合多尺度系统、带内部层的二维Burgers方程及三维时变问题。
英文摘要
Physics-informed neural networks (PINNs) have emerged as a meshless alternative to conventional numerical methods for solving partial differential equations (PDEs). However, their limited ability to capture sharp gradients can lead to substantial errors when resolving boundary and interior layers. Here, we introduce an energy-norm-enhanced PINN (ENPINN) that incorporates gradient information and variational structure into the loss function to improve the resolution of layer-dominated solutions. We first examine two related formulations: weak-loss PINNs (WLPINNs), which incorporate test functions into the conventional PINN residual, and gradient-enhanced PINNs (gPINNs), which augment the loss with spatial derivatives of the PDE residual. By analyzing these formulations, we identify their limitations in resolving steep solution gradients and motivate the systematic construction of ENPINN. We establish theoretically how the energy-norm error depends on the ENPINN loss and show that a suitably modified residual-derivative term is essential for accurately capturing boundary layers. We further establish the existence of neural-network approximations with arbitrarily small energy error and derive corresponding derivative bounds, providing a theoretical foundation for the proposed framework. The performance of ENPINN is assessed through systematic comparisons with existing PINN variants for convection-diffusion-reaction problems exhibiting steep gradients. Numerical experiments include a combustion model, a coupled multi-scale system, a two-dimensional Burgers equation with an interior layer, and a three-dimensional time-dependent problem.
Comments35 pages, 8 figures
