# Data assimilation and parameter identification for water waves using the nonlinear Schr"odinger equation and physics-informed neural networks

arXiv: https://arxiv.org/abs/2401.03708 · HTML: https://mlsanity.com/paper/2401.03708/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 80 references that are in the map.
- First posted on arXiv 8 Jan 2024.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks](https://mlsanity.com/paper/2005.03596/) (2020)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?](https://mlsanity.com/paper/2103.09655/) (2021)
- [A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications](https://mlsanity.com/paper/2010.02011/) (2020)
- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [Data-driven rogue waves and parameter discovery in the defocusing NLS equation with a potential using the PINN deep learning](https://mlsanity.com/paper/2012.09984/) (2020)
- [Physics-informed Neural Network for Nonlinear Dynamics in Fiber Optics](https://mlsanity.com/paper/2109.00526/) (2021)
- [Surface Similarity Parameter: A New Machine Learning Loss Metric for Oscillatory Spatio-Temporal Data](https://mlsanity.com/paper/2204.06843/) (2022)
- [Machine learning for phase-resolved reconstruction of nonlinear ocean wave surface elevations from sparse remote sensing data](https://mlsanity.com/paper/2305.11913/) (2023)

## Built on by (most cited first, 1 of 1)

- [Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force method](https://mlsanity.com/paper/2505.04875/) (2025)

## Abstract (arXiv)

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"odinger 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.