## 5 Discussion
The configuration of FP-GENN used here aims at minimizing the difference between the true anomaly state of the system \(d\mathbf{x}\) and its representation \(\psi(d\mathbf{x})\) through energy form \(||d\mathbf{x}-\psi(d\mathbf{x})||^{2}\). Alternate energy forms have to be investigated, considering extremes or more generally the whole pdf. In addition, the fixed-point solver used in the joint interpolation approach with GENN never goes too far from the observations, even though they are noisy, which can be an issue in cases of strong noise, including spatial and/or temporal correlations, which was already seen when using SWOT data without any preprocessing (not shown here). From a methodological point of view, the next developments are expected in the coming related works to increase the gain already observed with FP-GENN:  
* Use a joint learning of the dynamical representation \(\psi\) and the solver \(\Gamma\), minimizing its reconstruction error. A significant gain in the reconstruction performance is expected according to preliminary results obtained with toy models [16]. * A stochastic extension of GENN for including in the NN-based framework an estimation of the uncertainties, thereby enabling this new reconstruction method to fully compete with the other interpolators in a \"data assimilation\" context, with a possible link whith Gaussian processes and the related stochastic PDE formalism [18; 19]. Besides methodological aspects, new applications are also promising. If we focus here on small North Atlantic subdomains, the transfer of the NN-based interpolators to an operational process chain would mean reproducing a similar work on the whole basin wherein the computational constraints in this learning-based setting with a large number of parameters would still be a challenge. Using a deep learning multi-GPU framework and building a pre-operational demonstrator should be of great interest in the community, as are other SWOT use cases, e.g., using pre-learning on SWOT data to produce a new interpolation of historical along-track nadir datasets, or taking advantage of the SWOT fast-sampling phase data as inputs for learning prior to its use with SWOT upcoming \"operational\" data. Last, because the 2D information brought by SWOT showed a significant gain in the reconstruction, a natural extension of this work would be to consider pseudo-observations SKIM datasets [20], whose swath width is more than twice larger (110 vs. 270 km), and another would be to propose multivariate analyses, including complementary datasets (SST/SSS), already existing in other data-driven schemes such as AnDA, with an easy extension as additional channels in a neural networks framework. The code is available on [[https://github.com/CIA-Oceanix/DINAE_keras](https://github.com/CIA-Oceanix/DINAE_keras)]([https://github.com/CIA-Oceanix/DINAE_keras](https://github.com/CIA-Oceanix/DINAE_keras)) with additional information provided in the ReadMe file to describe the architecture of the code and how to use it. Conceptualization, R.F. and M.B. ([PERSON]); methodology, R.F. and M.B. ([PERSON]); software, R.F. and M.B. ([PERSON]); validation, M.B. ([PERSON]); investigation, M.B. ([PERSON]), R.F., C.U., M.B. ([PERSON]) and B.C. ; writing--original draft preparation, M.B. ([PERSON]). All authors have read and agreed to the published version of the manuscript. Funding Funding for the authors was provided by CNES (French Space Agency) through OSTST project MANATEE and SWOT ST project DIEGO. This work was also supported by ANR through project Melody and AI Chair OceaniX. It also benefited from HPC and GPU resources from Azure (Microsoft EU Ocean awards) and from GENCI-IDRIS (Grant No. 2020-101030). Special thanks to the French ANR project BOOST-SWOT for providing the datasets used in this work. The authors declare no conflict of interest.