# Machine learning for phase-resolved reconstruction of nonlinear ocean wave surface elevations from sparse remote sensing data

arXiv: https://arxiv.org/abs/2305.11913 · HTML: https://mlsanity.com/paper/2305.11913/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 82 references that are in the map.
- First posted on arXiv 18 May 2023.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
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- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Context Encoders: Feature Learning by Inpainting](https://mlsanity.com/paper/1604.07379/) (2016)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Model Evaluation, Model Selection, and Algorithm Selection in Machine Learning](https://mlsanity.com/paper/1811.12808/) (2018)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Seismic wave propagation and inversion with Neural Operators](https://mlsanity.com/paper/2108.05421/) (2021)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [When Image Denoising Meets High-Level Vision Tasks: A Deep Learning Approach](https://mlsanity.com/paper/1706.04284/) (2017)
- [A Robust Deep Learning Workflow to Predict Multiphase Flow Behavior during Geological CO2 Sequestration Injection and Post-Injection Periods](https://mlsanity.com/paper/2107.07274/) (2021)
- [Surface Similarity Parameter: A New Machine Learning Loss Metric for Oscillatory Spatio-Temporal Data](https://mlsanity.com/paper/2204.06843/) (2022)

## Built on by (most cited first, 2 of 2)

- [Data assimilation and parameter identification for water waves using the nonlinear Schr"odinger equation and physics-informed neural networks](https://mlsanity.com/paper/2401.03708/) (2024)
- [Bridging ocean wave physics and deep learning: Physics-informed neural operators for nonlinear wavefield reconstruction in real-time](https://mlsanity.com/paper/2508.03315/) (2025)

## Abstract (arXiv)

Accurate short-term predictions of phase-resolved water wave conditions are crucial for decision-making in ocean engineering. However, the initialization of remote-sensing-based wave prediction models first requires a reconstruction of wave surfaces from sparse measurements like radar. Existing reconstruction methods either rely on computationally intensive optimization procedures or simplistic modelling assumptions that compromise the real-time capability or accuracy of the subsequent prediction process. We therefore address these issues by proposing a novel approach for phase-resolved wave surface reconstruction using neural networks based on the U-Net and Fourier neural operator (FNO) architectures. Our approach utilizes synthetic yet highly realistic training data on uniform one-dimensional grids, that is generated by the high-order spectral method for wave simulation and a geometric radar modelling approach. The investigation reveals that both models deliver accurate wave reconstruction results and show good generalization for different sea states when trained with spatio-temporal radar data containing multiple historic radar snapshots in each input. Notably, the FNO demonstrates superior performance in handling the data structure imposed by wave physics due to its global approach to learn the mapping between input and output in Fourier space.