论文标题
在物理知识的神经网络和变异物理信息的神经网络中强制执行DIRICHLET边界条件
Enforcing Dirichlet boundary conditions in physics-informed neural networks and variational physics-informed neural networks
论文作者
论文摘要
在本文中，我们介绍并比较了四种方法，以在物理知识的神经网络（PINN）和变异物理信息信息中的神经网络（VPINN）中强制执行Dirichlet边界条件。这种条件通常是通过在损失函数中添加惩罚条款并正确选择相应缩放系数来施加的；但是，实际上，这需要一个昂贵的调整阶段。我们通过几个数值测试显示，这些测试修改神经网络的输出以与规定的值完全匹配，从而导致更有效，更准确的求解器。通过近似距离函数，通过精确执行Dirichlet边界条件来实现最佳结果。我们还表明，通过Nitsche的方法施加Dirichlet边界条件的变体导致次优求解器。
In this paper, we present and compare four methods to enforce Dirichlet boundary conditions in Physics-Informed Neural Networks (PINNs) and Variational Physics-Informed Neural Networks (VPINNs). Such conditions are usually imposed by adding penalization terms in the loss function and properly choosing the corresponding scaling coefficients; however, in practice, this requires an expensive tuning phase. We show through several numerical tests that modifying the output of the neural network to exactly match the prescribed values leads to more efficient and accurate solvers. The best results are achieved by exactly enforcing the Dirichlet boundary conditions by means of an approximate distance function. We also show that variationally imposing the Dirichlet boundary conditions via Nitsche's method leads to suboptimal solvers.
