The measurement of deep water gravity wave elevations using in-situ devices, such as wave gauges, typically yields spatially sparse data. This sparsity arises from the deployment of a limited number of gauges due to their installation effort and high operational costs. The reconstruction of the spatio-temporal extent of surface elevation poses an ill-posed data assimilation problem, challenging to solve with conventional numerical techniques. To address this issue, we propose the application of a physics-informed neural network (PINN), aiming to reconstruct physically consistent wave fields between two designated measurement locations several meters apart. Our method ensures this physical consistency by integrating residuals of the hydrodynamic nonlinear Schr\"{o}dinger equation (NLSE) into the PINN's loss function. Using synthetic wave elevation time series from distinct locations within a wave tank, we initially achieve successful reconstruction quality by employing constant, predetermined NLSE coefficients. However, the reconstruction quality is further improved by introducing NLSE coefficients as additional identifiable variables during PINN training. The results not only showcase a technically relevant application of the PINN method but also represent a pioneering step towards improving the initialization of deterministic wave prediction methods.

翻译：利用波高仪等原位设备测量深水重力波波面高度时，通常只能获得空间稀疏数据。这种稀疏性源于设备安装难度大、运行成本高导致布设数量有限。重构波面高度的时空分布是一个不适定的数据同化问题，传统数值方法难以求解。针对该问题，本文提出应用物理信息神经网络（PINN），旨在重构相距数米的两个指定测点间物理一致的波浪场。该方法通过将水动力学非线性薛定谔方程（NLSE）残差纳入PINN损失函数来保证物理一致性。利用波浪水槽中不同位置处的合成波面高度时间序列，我们通过使用恒定预设的NLSE系数初步实现了高质量重构。然而，通过在PINN训练过程中将NLSE系数作为额外可辨识变量引入，重构质量得到进一步提升。研究结果不仅展示了PINN方法的技术相关应用，更是向改进确定性波浪预报方法初始化迈出的开创性一步。