## 5 Discussion  
In this study focusing on how data-driven and learning-based algorithms may help to improve the reconstruction performances of altimetric fields generally given by a state-of-the-art optimal interpolation (OI) baseline, provided through the DUACS processing chain, we used two small areas with different energetic dynamics: the \(10^{\circ}\times 10^{\circ}\) GULFSTREAM domain mainly driven by mesoscale processes and the \(8^{\circ}\times 10^{\circ}\) OSMOSIS domain, less energetic but labeled with more small scale structures. Based on the NATL60 numerical simulations [8], some experiments were designed in which pseudo observational along-track nadir and wide-swath SWOT realistic datasets are generated. As the DUACS OI [7] of these pseudo-observations is used as the reference, all the investigated methods are applied in a multi-scale decomposition framework where the anomaly \(d\mathbf{x}\) is seen as the difference between the original field \(\mathbf{x}\) and the large-scale component \(\overline{\mathbf{x}}\) provided by the OI. Knowing the underlying reality, it was possible to precisely assess the reconstruction abilities of both AnDA and DINEOF data-driven methodologies, already consolidated with numerous experiences and methodological developments reported in the literature [2; 5; 6; 14]. As a new competitive learning-based approach, we proposed to apply specifically interpolation-designed neural networks involving a joint interpolation and representation learning for irregularly-sampled satellite-derived geophysical fields [15]. As a short synthesis of these evaluations reported in Sections 4.2 and 4.3, some key points can be retrieved:  
Figure 15: Global SSH gradient field reconstruction obtained by OI, (post-)AnDA, VE-DINEOF, FP-ConvAE and FP-GENN for a joint assimilation/learning of along-track nadir with wide-swath SWOT data. (a) Ground truth (\(\
abla_{\text{SSH}}\)), (b) OI, (c) post-AnDA, (d) VE-DINEOF, (e) FP-ConvAE, (f) FP-GENN. * A significant gain from data-driven methods compared to the OI-based DUACS baseline: up to 40% relative gain in the SSH daily root mean squared error, in particular on the GULFSTREAM domain where the small scale spatial patterns structures are less noticeable compared to OSMOSIS. * A better reconstruction performance of the learning-based GENN introducing a GMRF representation closely related to Gibbs energy concepts compared to AnDA and DINEOF. * A significant contribution from the 2D spatial information provided by the additional SWOT sampling to improve the reconstruction of altimetric fields with a relative gain up to 30% in the SSH daily mean squared error, when compared to the single use of along-track nadir 1D information. Within this combined use of the two datasets, the spectral analysis indicates the new capability to reconstruct spatial scales up to 50-60 km which is an important improvement compared to the scales that OI is handling by now; on the other hand, the temporal sampling being less important than nadir tracks, in particular on the GULFSTREAM domain where periods of several days without any SWOT information appears, the reconstruction on these specific periods is sometimes better when learning only with along-track nadir as inputs: we believe that a longer training period (not available here) should improve the behavior of the NN on this specific issue. * The possibility for neural network methods to learn from the single observations, without requiring any numerical simulations, which is of particular interest on low latitude areas where the Rossby radius of deformation is large, thereby requesting an important catalog to efficiently retrieve the SSH dynamics over the year. As it stands, the results obtained are very encouraging: FP-GENN is a \"plug-and-play\" algorithm whose conceptual use easily enables its implementation on new datasets. Many perspectives have to be considered in the short and medium terms.