A trial solution for imposing boundary conditions of partial differential equations in physics-informed neural networks

This article proposes an auxiliary function for imposing the boundary and initial conditions in physics-informed neural network models in a hard manner that accelerates the learning process. This auxiliary function consists of two pre-trained neural networks and a main deep neural network with trainable parameters. The novelty of this new auxiliary function is the input of main deep neural network, which takes the outputs of distance function and boundary function as inputs in addition to the spatiotemporal variables. We demonstrate the efficacy and general applicability of the proposed model by applying it to several benchmark-forward problems namely, advection, Helmholtz and Klein-Gordon equations. The accuracy of predictions is examined by comparison with exact solutions. Our findings imply the superiority of the proposed model because of the improvement of the loss convergence to lower values by one order of magnitude for the same number of epochs. In the case of the advection equation, the relative L2 error has been reduced from 0.025 to 0.0201, and from 0.016 to 0.0152. When applied to the Helmholtz equation, our novel model achieved an error of 8.67 × 10⁻³, surpassing the conventional model's performance, which yielded an error of 6.04 × 10⁻². Furthermore, for the Klein-Gordon equation, our new model led to a remarkable reduction in the relative L2 error, from 0.18 to an impressive 4.2 × 10⁻².