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In this paper, we present and compare four methods to enforce Dirichletboundary conditions in Physics-Informed Neural Networks (PINNs) and VariationalPhysics-Informed Neural Networks (VPINNs). Such conditions are usually imposedby adding penalization terms in the loss function and properly choosing thecorresponding scaling coefficients; however, in practice, this requires anexpensive tuning phase. We show through several numerical tests that modifyingthe output of the neural network to exactly match the prescribed values leadsto more efficient and accurate solvers. The best results are achieved byexactly enforcing the Dirichlet boundary conditions by means of an approximatedistance function. We also show that variationally imposing the Dirichletboundary conditions via Nitsche's method leads to suboptimal solvers.

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