**Abstract** : In Machine Learning (generally devoted to big-data case), the predictive learning (or the supervised learning) approach consists in identifying/learning a random mapping F: w↦ q = F(w), in which the parameters vector w (input) is modelled by a random vector W with known probability distribution Pw(dw) and where the vector of quantities of interest q (outputs) is the non-Gaussian random variable Q = F(W) = f(W,U) whose probability distribution is unknown, given an initial dataset (or training set) DN = {(wj,qj), j=1,…N} of N independent realizations of random vector (W,Q). The measurable mapping f is deterministic and U is a random vector whose probability distribution is known. The approach of probabilistic learning on manifold (recently introduced) will be presented, which allows for constructing a generator of an estimation of the joint probability distribution PW,Q(dw,dq; N) using only DN, which completely characterizes random mapping F. In this framework, novel computational statistical tools will be presented for the small-data challenge for which N is relatively small and consequently, is not sufficient large for constructing converged statistical estimates. In particular, we will present (1) the identification of the optimal independent component partition of the non-Gaussian random vector, (2) the learning from DN for which additional information is available based either on a nonparametric Bayesian approach or on Information Theory. Several applications will be presented such as, the identification of non-Gaussian random fields for random media, the Bayes inference with probabilistic learning, the robust design of an implant in a biological tissue at mesoscale, the nonparametric model-form uncertainties (i) in nonlinear solid dynamics applied a to MEMS and (ii) in nonlinear computational fluid dynamics applied to a Scramjet, nonconvex optimization under uncertainties.
[1] C. Soize, R. Ghanem, Data-driven probability concentration and sampling on manifold, Journal of Computational Physics, 321, 242-258 (2016).
[2] C. Soize, Optimal partition in terms of independent random vectors of any non-Gaussian vector defined by a set of realizations, SIAM/ASA Journal on Uncertainty Quantification, 5(1), 176-211 (2017).
[3] R. Ghanem, C. Soize, Probabilistic nonconvex constrained optimization with fixed number of function evaluations, International Journal for Numerical Methods in Engineering, 113(4), 719-741 (2018).
[4] C. Soize, Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm, Computational Mechanics, 62(3), 477-497 (2018).
[5] C. Soize, C. Farhat, Probabilistic learning for model-form uncertainties in nonlinear computational mechanics, International Journal for Numerical Methods in Engineering, Accepted 24 October 2018.
[6] C. Soize, R. Ghanem, C. Safta, X. Huan, Z.P. Vane, J. Oefelein, G. Lacaze, H.N. Najm, Enhancing model predictability for a scramjet using probabilistic learning on manifold, AIAA Journal, Accepted 13 September 2018.