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

S Berrone; C Canuto; M Pintore; N Sukumar

doi:10.1016/j.heliyon.2023.e18820

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

Heliyon. 2023 Aug 2;9(8):e18820. doi: 10.1016/j.heliyon.2023.e18820. eCollection 2023 Aug.

Authors

S Berrone¹, C Canuto¹, M Pintore¹, N Sukumar²

Affiliations

¹ Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
² Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, USA.

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.

Keywords: 35A15; 65K10; 65L10; 65L20; 68T05; Approximate distance function; Deep neural networks; Dirichlet boundary conditions; PINN; VPINN.