Deep Learning-Based Solution for the KdV-family Governing Equations of Ocean Internal Waves

Internal Solitary Waves (ISWs) are critical for ocean studies due to their large amplitude and long-travel capabilities. Conventionally, the Korteweg-de Vries (KdV) equations and their extensions are employed to simulate ISW properties, but traditional numerical methods lack flexibility and efficiency. This study introduces a universal, deep learning-based model that streamlines solving KdV-family equations. Within the framework of physics-informed neural networks, we implement an optimized Radial Basis Function (RBF) neural network and a new progressive expansion training strategy. This innovation minimizes error during training, leading to efficient convergence. Our model is tested on KdV and forced KdV equations, dimensional and non-dimensional equations using soliton, cnoidal, and dnoidal waveforms to simulate ISW propagation. The model results align with theoretical and numerical benchmarks, as demonstrated in a case study in the Sulu Sea. This paper does not concern ISW dynamics but uses the KdV equation as an example to showcase how to solve the partial differential equations with a new deep-learning method. The developed deep-learning model offers an efficient and accurate approach to solving KdV-family equations in oceanographic studies.