# Bridging ocean wave physics and deep learning: Physics-informed neural operators for nonlinear wavefield reconstruction in real-time

arXiv: https://arxiv.org/abs/2508.03315 · HTML: https://mlsanity.com/paper/2508.03315/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 64 references that are in the map.
- First posted on arXiv 5 Aug 2025.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [SoftAdapt: Techniques for Adaptive Loss Weighting of Neural Networks with Multi-Part Loss Functions](https://mlsanity.com/paper/1912.12355/) (2019)
- [Applications of physics informed neural operators](https://mlsanity.com/paper/2203.12634/) (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)
- [Physics-informed neural networks for phase-resolved data assimilation and prediction of nonlinear ocean waves](https://mlsanity.com/paper/2501.08430/) (2025)

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

- [Leveraging Differentiable PDE Solvers for Semi-Neural Spatial Reconstruction From Sparse Measurements](https://mlsanity.com/paper/2601.20496/) (2026)

## Abstract (arXiv)

Accurate real-time prediction of phase-resolved ocean wave fields remains a critical yet largely unsolved problem, primarily due to the absence of practical data assimilation methods for reconstructing initial conditions from sparse or indirect wave measurements. While recent advances in supervised deep learning have shown potential for this purpose, they require large labelled datasets of ground truth wave data, which are infeasible to obtain in real-world scenarios. To overcome this limitation, we propose a Physics-Informed Neural Operator (PINO) framework for reconstructing spatially and temporally phase-resolved, nonlinear ocean wave fields from sparse measurements, without the need for ground truth data during training. This is achieved by embedding residuals of the free surface boundary conditions of ocean gravity waves into the loss function of the PINO, constraining the solution space in a soft manner. After training, we validate our approach using highly realistic synthetic wave data and demonstrate the accurate reconstruction of nonlinear wave fields from both buoy time series and radar snapshots. Our results indicate that PINOs enable accurate, real-time reconstruction and generalize robustly across a wide range of wave conditions, thereby paving the way for operational, data-driven wave reconstruction and prediction in realistic marine environments.