This is the situation when a model is accurate enough to describe the subjacent physics almost everywhere, except in a small region where the reality differs from the assumed model. This is the case when, in solid mechanics, some localized damage occurs, which involves a degradation of the real mechanical properties with respect to the ones assumed in the so-called nominal model. 2. When a model constitutes an approximation of the real system everywhere but is not accurate enough everywhere. This is the case when modeling large structures, where the discretization remains poor for reflecting all structural details. Thus, all these details (stiffeners, holes, etc.) are ignored and the resulting model could constitute an approximation (sometimes a crude approximation) of the real system, which is valuable for certain applications but inefficient in other cases. In the first case,we can talk of damage detection, and over the last decades, many techniques have been developed [9]. In particular, it is possible to classify the general used methods in frequency and time domains. Frequency domain techniques [10; 11; 12; 13; 14] are based on modal analysis, and this can be represent a great advantage when a range of frequencies to be analysed is known a priori. Time domain techniques [15; 16; 17; 18; 19; 20] are, instead, based on the study of vibration features and signal processing techniques. Wavelet analysis [21; 22; 23; 24; 25] is another used technique that uses a wavelet transformation on the modal shape of vibration. Due to its nature, these techniques are largely used for non-destructive testing. Moreover, during the last years, the interest to refine finite element models has grown, and an extensive review is provided in [26; 27]. This paper aims to propose a new methodology able to complete the data acquired from few sensors, as well as to find the model's correction (a sort of model reparation). This correction will be used to identify the damage localization and its severity. For that purpose, sparse identification will be the main protagonist within an hybrid approach combining the nominal model with the requested model enrichment, which is the last learned from the assimilated data. The second scenario is even more challenging due to the fact that the model should be corrected (enriched) everywhere and, in general, only a few data available. In this case, since the enrichment does not exhibit localization, the sparse identification will be replaced by global identifications. The next section describes the main methodologies that will be used in the proposed data completion and model enrichment procedures, particularly sparse regularization and identification. Section 3 will focus in the main paper's proposal, the procedures for completing the data and enriching the nominal models in both cases when the enrichment exhibits, or not, localization. Finally, Section 4 will present and discuss some numerical results. In particular, for the local correction procedure, three different geometries (simple plate in Section 4.1, plate with hole in Section 4.2 and an L-shape structure in Section 4.3) are used to show the generality and applicability of the methodology. On the other hand, we only focus on one problem for global corrections (simple plate in Section 4.4).