Data Completion, Model Correction and Enrichment Based on Sparse Identification and Data Assimilation  
[PERSON]  
1 ESI Group, 3 bis Rue Saarinen, CEDEX, 94528 Rungis, France 12 PIMM, CNRS, ENSAM, HESAM, 151 Boulevard de l'Hopital, 75013 Paris, France 2  
[PERSON]  
2 PIMM, CNRS, ENSAM, HESAM, 151 Boulevard de l'Hopital, 75013 Paris, France 23 ESI Group Chair CREATE-ID, ENSAM Institute of Technology, 151 Boulevard de l'Hopital, 75013 Paris, France 3  
[PERSON]  
4 Instituto Tecnologico de Santo Domingo, Av. Los Proceres, Jardines del Norte, Santo Domingo 10602, Dominican Republic; [EMAIL_ADDRESS] 4  
[PERSON]  
5 Aragon Institute  
nonlinear cases, many algebraic systems must be generally solved in the former case for computing the time evolution of the searched solution and in the last case for ensuring the convergence of the non-linear solution procedure, which operates by linearization. All this is currently very well known, and it is performed every day in many routine applications of numerical simulation. However, when fast and accurate solutions becomes compulsory, three main difficulties remain open:  
1. The computational costs associated with the solution of those algebraic problems of size N, when N becomes large, e.g., several millions. In that case, the real-time feedback compulsory in most of the very timely digital applications is compromised. The use of larger and more powerful computing platforms is not the most appealing alternative due to the fact that online applications in autonomous systems need to embed these solvers in deployed systems in order to monitor the systems for real-time decision-making, based on efficient diagnosis and prognosis. 2. The second difficulty concerns the underlying model's quality. In some cases, the existing and experienced models are very accurate; however, in many other cases, as the ones concerning very large and complex systems, involving many coupled physical phenomena, seriously compromised the validity of predictions performed with the established state-of-the-art models. 3. The third difficulty concerns the efficient treatment of parametric models, for which its solutions will depend on the parameters involved in the problem's formulation. Sometimes these parameters concerns the physical model itself; other times, they affect the geometry of the domain in which the problems are defined; finally, at other times, these parameters are related to the loading (in the most general sense) that the system is experiencing. In all cases, when addressing parametric models, the curse of dimensionality (combinatorial explosion) joins the scene. Calculating the parametric solution of the problem could be an excellent opportunity for real-time engineering: for instance, solution particularization, optimization, inverse analysis, uncertainty propagation, and simulation-based control; all are performed under the stringent real-time constraint. However, the curse of dimensionality makes the calculation of those parametric solutions difficult when the number of parameters that they involve increases. In this general framework, the first issue can be addressed with Model Order Reduction (MOR) techniques [1; 2; 3; 4]. The second one can be addressed within an hybrid engineering paradigm, where collected and assimilated data can be used for enriching state-of-the-art models [5; 6]. The third issue can be addressed in the context of advanced regressions that are able to address richness (nonlinearities) in multi-parametric settings (many parameters), keeping the amount of required data reasonable and avoiding overfitting phenomena [7; 8]. The present work will focus specifically on the second difficulty to address two types of scenarios:  
1. When a model, called a nominal model, is expected to predict the system's response from the applied loading but fails to predict the measurements carried out due to a localized deficiency in it.