Boundary conditions for physics-informed neural networks applied to shallow-water equations

Physics-informed neural networks (PINNs) are neural networks that incorporate governing equations into their loss functions and satisfy these equations by minimizing the loss functions. This study investigated the applicability of PINNs in analyzing boundary conditions for two-dimensional shallow-water equations (SWEs). The PINNs developed in this study incorporated SWEs as well as initial and boundary conditions into the loss function. They were trained and evaluated under several types of boundaries, including completely reflective, nonreflective, partially reflective, wave-incident, and nonrestricted boundaries. The results of the PINNs were validated against the reference solution of the finite-difference method, demonstrating that the PINNs can accurately simulate these boundary conditions. When the boundary was undefined in the loss function (i.e. a nonrestricted boundary), it exhibited behavior similar to that of a nonreflective boundary, with a moderately large variance among the trials. Furthermore, when the nonrestricted boundary intersected with a completely reflective boundary, the PINNs behaved as if the completely reflective boundary extended beyond the computational domain.