BTF-PINN：无需边界训练的狄利克雷边界条件强制方法
BTF-PINN: Enforcing Dirichlet Boundary Conditions Without Boundary Training
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中文总结 AI 辅助
本文提出BTF-PINN这一无边界训练的物理信息神经网络，通过嵌入无边界损失函数求解齐次狄利克雷边值问题，经数值实验验证其有效性及更优的边界迹线精度。
中文摘要 AI 辅助
齐次狄利克雷边值问题体现了用非插值方法求解狄利克雷问题的核心难点。本文提出BTF-PINN（无边界训练的物理信息神经网络），这是一种仅在内部求解齐次狄利克雷边值问题的策略，无需边界训练、边界惩罚或适配边界的参数化。核心思想是将本质边界条件嵌入新设计的无边界损失函数中。我们证明了所提出的仅内部变分形式与原边值问题等价，建立了残差权重的尖锐阈值条件，并基于泛函的强制性开展收敛分析。对高维问题、不规则几何及各向异性椭圆方程的数值实验验证了BTF-PINN的有效性，与标准带边界惩罚的PINNs对比进一步表明，BTF-PINN实现了更优的边界迹线精度。
英文摘要
The homogeneous Dirichlet boundary value problem captures the core difficulty of solving Dirichlet problems with non-interpolatory methods. We propose BTF-PINN (Boundary-Training-Free Physics-Informed Neural Network), an interior-only strategy for solving homogeneous Dirichlet boundary value problems that requires no boundary training, boundary penalties, or boundary-conforming parametrizations. The key idea is to embed the essential boundary condition into a newly designed boundary-free loss function. We prove the equivalence between the proposed interior-only variational formulation and the original boundary value problem, establish a sharp threshold condition for the residual weight, and develop a convergence analysis based on the coercivity of the functional. Numerical experiments on high-dimensional problems, irregular geometries, and anisotropic elliptic equations demonstrate the effectiveness of BTF-PINN. Comparisons with standard boundary-penalty PINNs further show that BTF-PINN achieves superior boundary trace accuracy.
