Berrone S, Canuto C, Pintore M, Sukumar N
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, USA.
Heliyon. 2023 Aug 2;9(8):e18820. doi: 10.1016/j.heliyon.2023.e18820. eCollection 2023 Aug.
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.
在本文中，我们提出并比较了四种在物理信息神经网络（PINNs）和变分物理信息神经网络（VPINNs）中施加狄利克雷边界条件的方法。通常通过在损失函数中添加惩罚项并适当选择相应的缩放系数来施加此类条件；然而，在实践中，这需要一个昂贵的调优阶段。我们通过几个数值测试表明，修改神经网络的输出以精确匹配规定值会得到更高效和准确的求解器。通过使用近似距离函数精确施加狄利克雷边界条件可获得最佳结果。我们还表明，通过尼茨方法变分施加狄利克雷边界条件会导致次优求解器。
