Exploring two-dimensional internal waves: A new three-coupled Davey–Stewartson system and physics-informed neural networks with weight assignment methods

In order to investigate various two-dimensional internal waves in a two-layer finite-water depth fluid with different velocity potentials in each layer, we derive a new (2+1)-dimensional system of three coupled, two-component Davey–Stewartson type equations under the long wave assumption involving complex amplitude and real velocity potentials in the upper and lower layers. It is found that exact solutions of this system can describe internal solitary waves, internal rogue waves, and internal breathers, and more interestingly, it is proposed for the first time that the patterns of changes in the velocity potentials of the upper and lower layers can be used as an indicator for recognizing the occurrence, disappearance, and amplitudes of these internal waves. In view of the application of deep learning techniques in the simulation, prediction and monitoring of internal waves, an improved physical information neural network (PINN) method is introduced to successfully simulate dynamics of internal solitary waves, internal rogue waves, and internal breathers, as well as the corresponding velocity potentials. The effects of no weight assignment and two different weight assignment methods in the loss function on the training results are analyzed in detail, which reveals that better simulating results can be achieved via a fixed-weight loss function for small-amplitude internal waves, while the introduction of weights has little effect on the simulations for larger-amplitude internal waves. In addition, the loss function with adaptive weights performs well in the fast simulation of internal rogue waves and poorly in the learning effect of the velocity potentials, which is contrary for internal solitary waves, but is not applicable in the simulation of internal breathers.