Physics-informed neural networks with hard and soft boundary conditions for linear free surface waves

Physics-informed neural networks (PINNs) are introduced to solve the linear wave problem described by potential flow theory. In the proposed PINN framework, both soft and hard enforcing of boundary conditions (BCs) at the bottom and sides of the wave domain are proposed. Two scenarios for solving the linear wave problem are investigated to find suitable PINN architectures. In the first scenario, the free surface wave is considered to be completely defined, and in the second scenario, the wave angular frequency is considered an unknown parameter. With a completely defined free surface wave and incorporating both the free surface and bottom BCs as soft constraints, the average velocity distribution error is less than 3%. Incorporation of a periodic BC (PBC) as a soft constraint reduces the average error to 0.10%. A hard constraint PBC gives an average error of 0.16%, with a strict representation of the PBC. This study also explores the design of trial functions to impose the kinematic bottom BC (KBBC) as a hard constraint. While these trial functions strictly satisfy the KBBC, they limit the solution space, increasing the average error up to almost 15 times. When the angular frequency of the wave is considered as an unknown parameter, to be estimated by the PINN, its value is estimated with an average error of 0.03%. Since linear wave theory underpins many wave simulation approaches, the results of this study lay the groundwork for extending the PINN framework to more complex wave models.