WaveNets: physics-informed neural networks for full-field recovery of rotational flow beneath large-amplitude periodic water waves

We formulate physics-informed neural networks (PINNs) for full-field reconstruction of rotational flow beneath nonlinear periodic water waves using a small amount of measurement data, coined WaveNets. The WaveNets have two NNs to, respectively, predict the water surface, and velocity/pressure fields. The Euler equation and other prior knowledge of the wave problem are included in WaveNets loss function. We also propose a novel method to dynamically update the sampling points in residual evaluation as the free surface is gradually formed during model training. High-fidelity data sets are obtained using the numerical continuation method which is able to solve nonlinear waves close to the largest height. Model training and validation results in cases of both one-layer and two-layer rotational flows show that WaveNets can reconstruct wave surface and flow field with few data either on the surface or in the flow. Accuracy in vorticity estimate can be improved by adding a redundant physical constraint according to the prior information on the vorticity distribution.