Enforcing Dirichlet boundary conditions in physics-informed neural networks and variational physics-informed neural networks

Abstract

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.

22 pages, 45 figures

References in corpus (4)

• Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
• Variational Physics Informed Neural Networks: the role of quadratures and test functions
• Adaptive deep density approximation for Fokker-Planck equations
• Solving PDEs by Variational Physics-Informed Neural Networks: an a posteriori error analysis

Cited by in corpus (4)

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• Optimizing Variational Physics-Informed Neural Networks Using Least Squares
• Meshfree Variational Physics Informed Neural Networks (MF-VPINN): an adaptive training strategy
• Unfitted finite element interpolated neural networks