用有限元法增强PINN，精准强制边界条件，提升复杂问题求解稳定性。
PINN-FEM: A Hybrid Approach for Enforcing Dirichlet Boundary Conditions in Physics-Informed Neural Networks
- 结合PINN与FEM，通过区域分解在边界附近强约束狄利克雷条件
- 六组实验显示精度和鲁棒性显著优于传统PINN，收敛更稳定
- 适合工业与科学计算中的复杂几何场景，通用性强
物理信息神经网络（PINNs）通过将控制方程及边界/初始条件嵌入损失函数来求解偏微分方程。然而，精确施加狄利克雷边界条件仍具挑战，常导致软约束，影响复杂域下的收敛性与可靠性。本文提出一种混合方法PINN-FEM，结合PINNs与有限元法（FEM），通过区域分解在边界附近引入FEM表示，实现强边界条件的精确强制，且不损害收敛性。六个逐步复杂的实验表明，PINN-FEM在准确性和鲁棒性上均优于标准PINN模型。尽管已有距离函数等方法用于边界条件强制，但其通用性不足。PINN-FEM通过在边界附近利用FEM，有效弥合此差距，适用于工业与科学计算中的真实问题。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks (PINNs) solve partial differential equations (PDEs) by embedding governing equations and boundary/initial conditions into the loss function. However, enforcing Dirichlet boundary conditions accurately remains challenging, often leading to soft enforcement that compromises convergence and reliability in complex domains. We propose a hybrid approach, PINN-FEM, which combines PINNs with finite element methods (FEM) to impose strong Dirichlet boundary conditions via domain decomposition. This method incorporates FEM-based representations near the boundary, ensuring exact enforcement without compromising convergence. Through six experiments of increasing complexity, PINN-FEM outperforms standard PINN models, showcasing superior accuracy and robustness. While distance functions and similar techniques have been proposed for boundary condition enforcement, they lack generality for real-world applications. PINN-FEM bridges this gap by leveraging FEM near boundaries, making it well-suited for industrial and scientific problems.
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