Project Summary

• Investigated whether adding knowledge of ocean physics to neural networks can improve the reconstruction of subsurface ocean temperature and salinity from sparse observations.
• Developed a physics-informed neural network based on a coordinate-based Implicit Neural Representation (INR) with a SIREN architecture. The model combines sparse observations from Argo floats and satellites with physical equations describing processes in the ocean.
• Tested nine oceanographic physical constraints individually and in combination using both artificially sparsified GLORYS12V1 reanalysis data and real Argo, OSTIA, and satellite salinity observations.
• Found that most of the tested physical constraints improved temperature and salinity reconstruction, with improvements of around 32–38% compared with the corresponding physics-free models. However, temperature and salinity conservation increased the error because the implemented equations did not account for heat and freshwater entering or leaving the ocean through the surface.
• Showed that the way physics is introduced during training is important for stable results. Starting from a pretrained neural network was substantially more stable than starting from random weights.

Development Tools

• PyTorch [9]: Main machine-learning framework used to develop and train the neural networks. Its automatic differentiation functionality was used to calculate the derivatives needed by the physical constraints.
• gsw-torch [4]: Differentiable PyTorch implementation of the TEOS-10 seawater equation of state [3]. It was used to calculate seawater density while keeping the calculation connected to the neural network for physics-informed training.
• xarray / zarr: Used to efficiently access and process the large GLORYS12V1 reanalysis dataset.
• Copernicus Marine Toolbox [10]: Used to access the GLORYS12V1 ocean reanalysis data.
• pandas / netCDF4: Used to process and organize observational datasets, including Argo/EN4, OSTIA, and satellite salinity data.
• DepthDif Toolkit [11]: Adapted to retrieve and combine satellite and in-situ observations for the study region.
• uv [12]: Used to manage Python dependencies and keep the development environment reproducible.

Development Outputs

• Project codebase: https://github.com/brunacandido/inr_pinn_ocean

Project Description

Reconstructing the ocean's three-dimensional structure from observations is challenging because measurements are not available everywhere. Satellites provide frequent and spatially extensive information about the ocean surface, but information below the surface is much more limited. Argo floats provide valuable temperature and salinity profiles, but they are relatively sparse and irregularly distributed.

Ocean reanalysis products such as GLORYS12V1 can provide a complete estimate of the ocean by combining observations with numerical ocean models. However, these products can be computationally expensive to produce and are not perfect representations of the real ocean. This project investigated whether machine learning could help reconstruct the subsurface ocean from sparse observations and whether explicitly incorporating knowledge of ocean physics could improve reconstruction accuracy.

The main question was whether a Physics-Informed Neural Network (PINN) [1] could outperform a standard data-driven neural network. Instead of asking the network to learn only from the available observations, the PINN is also encouraged to produce results that are consistent with known physical relationships in the ocean.

Study area: a 4°×4° patch in the south-western Indian Ocean, centered at 34°S, 69°E, used for all experiments in this project.

Approach

The model used in this project is an Implicit Neural Representation (INR) that takes coordinates as input. Instead of predicting the ocean state on a predefined grid, the network receives a location and time as input  (longitude, latitude, depth, time) and predicts the temperature and salinity at that point.

The INR uses a SIREN architecture [2], which is a neural network based on sinusoidal activation functions. One advantage of this approach is that it can work directly with irregularly distributed observations, such as measurements collected by drifting Argo floats, without first interpolating them onto a regular grid.

The physics-informed version of the model incorporates physical constraints into the standard data-fitting objective. During training, the network is evaluated at additional points where there may be no observations. At these points, the model is encouraged to satisfy equations that describe known ocean processes.

Automatic differentiation was used to calculate the derivatives required by these equations directly from the neural network. This allows the model to learn from both observations and physical relationships simultaneously.

A particularly important part of the implementation was the calculation of seawater density. Several of the oceanographic equations depend on density, which is calculated from temperature, salinity, pressure, and other properties using the TEOS-10 standard [3]. The commonly used `gsw` library could not be used directly for this purpose because its calculations cannot be differentiated by PyTorch. Therefore, a differentiable PyTorch implementation, `gsw-torch` [4], was used to keep density calculations connected to the neural network during training.

Nine physical constraints were implemented:

• Equation of state
• Hydrostatic balance
• Mass continuity
• Geostrophic balance
• Thermal-wind balance
• Quasi-geostrophic potential vorticity
• Static stability
• Salinity conservation
• Temperature conservation

Each constraint was first tested separately to understand which physical relationships were most useful. Different combinations were then tested to investigate whether using several physical constraints together could yield further improvements.

Study Design

The experiments were carried out over a 4° × 4° region in the south-western Indian Ocean.

Two main types of experiments were performed.

First, GLORYS12V1 reanalysis data [5] were used. The data were artificially made sparse by removing between 50% and 99% of the available subsurface observations. This provided a controlled environment where the original GLORYS12V1 field could still be used as a reference for evaluating the reconstruction.

The second type of experiment used real observations. These included Argo/EN4 [6, 7] temperature and salinity profiles, OSTIA [8] sea-surface temperature, and satellite-derived sea-surface salinity. These experiments were used to test whether the findings from the controlled GLORYS12V1 experiments also applied to real-world observations.

For every physics-informed model, a corresponding physics-free INR was trained for the same amount of time. This was important because simply training a model for longer can sometimes improve its results. Comparing the two models at the same training stage made it possible to better isolate the effect of adding the physical constraint.

Key Findings

The results showed that adding physical information can significantly improve the reconstruction of the ocean, but that the choice and formulation of the physical constraints are important.

Most of the tested physical constraints improved both temperature and salinity reconstruction compared with their corresponding physics-free models. The improvements were generally 32–38% across most tested sparsity levels. However, the improvement became smaller at the highest level of sparsity. This was an unexpected result, since one might expect physical information to become increasingly useful as observations become more limited.

Effect of each physical constraint on temperature and salinity reconstruction error (PINN minus its own physics-free control), across data conditions from full data (E1/E2) to 99% subsurface sparsity. Green means the physics constraint helped; red means it hurt.

The temperature and salinity conservation constraints consistently worsened the reconstructions. The reason was identified in the formulation of these equations. The implemented conservation equations did not account for heat and freshwater entering or leaving the ocean through its surface. This is particularly important near the surface, where these processes are significant. This result shows that simply adding a physically meaningful equation does not guarantee better results: the equation must also represent the relevant physical processes in the region where it is applied.

Using several physical constraints together did not always produce better results than using a single constraint. Some combinations were beneficial, while others provided little improvement. The results, therefore, suggest that adding more equations is not necessarily better and that interactions among different physical constraints need to be considered.

The training strategy also had a strong effect on the results. Starting the physics-informed training from a previously trained INR was much more stable than starting from a randomly initialized network. Warm-starting reduced the extreme failures observed with random initialization and produced more consistent results across the different physical constraints. This suggests that the network's starting point can be as important as the choice of physical equation.

Warm-start vs. random initialization: change in validation temperature RMSE, by equation, relative to each arm's own epoch-matched control. Random initialization (epoch0) produces the most extreme outcomes in both directions; warm-starting is consistently the tightest column.

The same pattern held when the models were trained directly on real observations and evaluated against a much larger, independent Argo record spanning 26 years, confirming that the benefit of warm-starting is not limited to the controlled, synthetic experiments.

Real-observation-trained PINN family vs. real Argo, all years (102 profiles, 2000-2025): percentage change in temperature RMSE from the INR baseline, by equation and warm-start/weighting strategy. Green means the physics constraint helped; red means it hurt.

A comparison between models trained on synthetic GLORYS12V1 data and models trained directly on sparse real observations also showed that the synthetic-data approach could produce lower temperature errors when evaluated against Argo observations. However, this advantage became smaller and partly reversed when the models were evaluated against a larger, multi-year Argo dataset. This highlights the importance of testing reconstruction methods against independent and sufficiently large observational datasets.

RMSE against real Argo profiles, by depth, for a PINN trained on real observations vs. a PINN trained on synthetically sparsified GLORYS12V1, pooled across 102 real Argo profiles spanning 22 of the last 26 years. The real-observation model is more accurate near the surface; the synthetic-data model remains more accurate below roughly 750 m.

Overall, the collaboration with ESA Φ-lab resulted in a machine-learning framework that combines sparse Earth observation data with physical knowledge of the ocean. The project not only demonstrated that physics can improve neural-network-based ocean reconstruction, but also identified situations where physical constraints can be counterproductive and showed that training strategy and reference-data quality have a major influence on the results.

References

1. Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. *Journal of Computational Physics*, 378, 686-707.

2. Sitzmann, V., Martel, J. N. P., Bergman, A. W., Lindell, D. B., & Wetzstein, G. (2020). Implicit Neural Representations with Periodic Activation Functions. arXiv:2006.09661.

3. McDougall, T. J., & Barker, P. M. (2011). *Getting started with TEOS-10 and the Gibbs Seawater (GSW) Oceanographic Toolbox*. SCOR/IAPSO WG127.

4. Miranda, J. R., & Zavala-Romero, O. (2026). *GSW PyTorch: PyTorch Implementation of the Gibbs SeaWater Oceanographic Toolbox* (v0.1.0). https://github.com/0jrm/gsw-torch

5. Lellouche, J.-M., et al. (2021). The Copernicus Global 1/12° Oceanic and Sea Ice GLORYS12 Reanalysis. *Frontiers in Earth Science*, 9, 698876.

6. Johnson, G. C., et al. (2022). Argo—Two Decades: Global Oceanography, Revolutionized. *Annual Review of Marine Science*, 14, 379-403.

7. Good, S. A., Martin, M. J., & Rayner, N. A. (2013). EN4: Quality controlled ocean temperature and salinity profiles and monthly objective analyses with uncertainty estimates. *Journal of Geophysical Research: Oceans*, 118(12), 6704-6716.

8. Worsfold, M., Good, S., McLaren, A., Fiedler, E., Roberts-Jones, J., & Martin, M. (2023). *Global Ocean OSTIA sea surface temperature and sea ice reprocessed*.

9. Paszke, A., et al. (2019). PyTorch: An imperative style, high-performance deep learning library. *Advances in Neural Information Processing Systems*, 32.

10. Copernicus Marine Service (2026). *Copernicus Marine Toolbox*. https://github.com/mercator-ocean/copernicus-marine-toolbox

11. Donike, S., & Cartuyvels, R. (2026). *DepthDif: Ambient Diffusion for a Global Dense Reconstruction of Subsurface Ocean Variables from Extremely Sparse Inputs* (v0.1.0). https://github.com/simon-donike/DepthDif

12. Astral Software Inc. (2024). *uv: An extremely fast Python package and project manager*. https://github.com/astral-sh/uv