Physics-Informed Neural Networks with Two Weighted Loss Function Methods for Interactions of Two-Dimensional Oceanic Internal Solitary Waves

The multiple patterns of internal solitary wave interactions (ISWI) are a complex oceanic phenomenon. Satellite remote sensing techniques indirectly detect these ISWI, but do not provide information on their detailed structure and dynamics. Recently, the authors considered a three-layer fluid with shear flow and developed a (2+1) Kadomtsev-Petviashvili (KP) model that is capable of describing five types of oceanic ISWI, including O-type, P-type, TO-type, TP-type, and Y-shaped. Deep learning models, particularly physics-informed neural networks (PINN), are widely used in the field of fluids and internal solitary waves. However, the authors find that the amplitude of internal solitary waves is much smaller than the wavelength and the ISWI occur at relatively large spatial scales, and these characteristics lead to an imbalance in the loss function of the PINN model. To solve this problem, the authors introduce two weighted loss function methods, the fixed weighing and the adaptive weighting methods, to improve the PINN model. This successfully simulated the detailed structure and dynamics of ISWI, with simulation results corresponding to the satellite images. In particular, the adaptive weighting method can automatically update the weights of different terms in the loss function and outperforms the fixed weighting method in terms of generalization ability.