用神经网络+特殊测试函数，精准求解含边界层的微分方程。
A Variational Physics-Informed Neural Network Framework Using Petrov-Galerkin Method for Solving Singularly Perturbed Boundary Value Problems
- 用神经网络做试函数，帽函数做测试函数，结合弱形式求解。
- 在L2和最大范数下精度显著优于标准VPINN方法。
- 适合求解含小参数、有边界层的微分方程，如传热或流体问题。
本文提出一种变分物理信息神经网络（VPINN）框架，将佩特罗夫-加勒金公式与深度神经网络结合，用于求解一维奇异摄动边值问题（BVPs）及含一个或两个小参数的抛物型偏微分方程。该方法采用非线性逼近：试函数空间由神经网络函数定义，测试函数空间由帽函数构建。通过局部化测试函数构造弱形式，并引入界面罚项以增强数值稳定性并精确捕捉边界层。狄利克雷边界条件通过硬约束施加，源项利用自动微分计算。在基准问题上的数值实验表明，该方法对一维奇异摄动微分方程的求解，在L2范数和最大范数下的精度均显著优于标准VPINN方法。
原文摘要 · Abstract (English)
This work proposes a Variational Physics-Informed Neural Network (VPINN) framework that integrates the Petrov-Galerkin formulation with deep neural networks (DNNs) for solving one-dimensional singularly perturbed boundary value problems (BVPs) and parabolic partial differential equations (PDEs) involving one or two small parameters. The method adopts a nonlinear approximation in which the trial space is defined by neural network functions, while the test space is constructed from hat functions. The weak formulation is constructed using localized test functions, with interface penalty terms introduced to enhance numerical stability and accurately capture boundary layers. Dirichlet boundary conditions are imposed via hard constraints, and source terms are computed using automatic differentiation. Numerical experiments on benchmark problems demonstrate the effectiveness of the proposed method, showing significantly improved accuracy in both the $L_2$ and maximum norms compared to the standard VPINN approach for one-dimensional singularly perturbed differential equations (SPDEs).
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