We propose real-time approximation methods for multi-parametric static nonlinear mechanical problems. Data assimilation is based on high-fidelity computations provided by a solver in a black-box routine. We suppose that there exists an approximation that accurately captures the data in the parameter space and that it has low rank intrinsic structure. In [1] the proposed Sparse Subspace Learning (SSL) method is a non-intrusive hierarchical collocation framework coupled with an incremental low-rank approximation (iRSVD). But, the Lagrange interpolation and the constrained collocation coupled to the hierarchical algorithm lead to oscillations and error issues. Therefore, two different approaches are proposed in order to circumvent those. Firstly, iRSVD is coupled to a regression based on Sobolev norm (MSN). This makes it possible to remove Runge oscillation when too many samples and parameters are used. Secondly, we couple rank revealing randomized SVD [2] and the sparsity-promoting regression framework SINDy [3]. We seek to build a library of multidimensional basis functions and then identify by hard thresholding the most pertinent coefficients in the regression. Our method generalizes the construction of the basis library in an incremental manner using Smolyak’s rule. The aim is to mitigate the curse of dimensionality we are subjected to when building the library in high dimensions. In both methods, sampling is considered fully random in the parametric domain. Specific strategies, and incremental procedures make it possible to reduce the number of high fidelity computations and ensure a global tolerance for the reduced order models. Hence we obtain a real time interactive software. Studies of academic and realistic models are proposed with material, process or load parameter variations. REFERENCES [1] Borzacchiello, D., Aguado, J.V. & Chinesta, F. Non-intrusive Sparse Subspace Learning for Parametrized Problems. Arch Computat Methods Eng, 26, 303–326, 2019. [2] A Rank Revealing Randomized Singular Value Decomposition (R3SVD) Algorithm for Low-rank Matrix Approximations, H. Ji, W. Yu, Y. Li, arXiv:1605.08134 [cs.NA], 2016. [3] Brunton SL, Proctor JL, Kutz JN. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. 113, 3932–3937, 2016.