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.

我们制定了物理信息神经网络（PINN），使用少量测量数据对非线性周期性水波下的旋转流进行全场重建，创造了 WaveNet。WaveNet 有两个神经网络，分别用于预测水面和速度/压力场。WaveNets 损失函数中包含了欧拉方程和其他波动问题的先验知识。我们还提出了一种新方法，随着模型训练过程中自由表面的逐渐形成，动态更新残差评估中的采样点。使用数值连续方法获得高保真数据集，该方法能够求解接近最大高度的非线性波。一层和两层旋转流情况下的模型训练和验证结果表明，WaveNets 可以用很少的表面或流中数据重建波面和流场。根据涡度分布的先验信息，通过添加冗余物理约束，可以提高涡度估计的精度。