Data-driven probabilistic learning on manifolds
Résumé
A novel methodology (Soize & Ghanem, JCP 2016} allows for generating realizations of a vector-valued random variable X whose probability distribution is unknown. The solely available information is a given dataset of independent realizations of X, represented by a matrix. The probabilistic learning methodology proposed is based (1) on the construction of a diffusion-maps basis (Coifman 2005) , which allows for characterizing the local geometry structure of data set; (2) on the construction of a Reduced Itô Stochastic Differential Equation (R-ISDE) whose invariant measure is constructed as the projection of the ISDE on the diffusion maps basis, associated with a dissipative Hamiltonian dynamical system (Soize 2008, Soize & Ghanem, JCP 2016) for which the invariant measure is given and estimated with the dataset. The MCMC generator of realizations is given by solving the R-ISDE with a Stormer-Verlet algorithm and we then show (Soize and Ghanem, JCP 2017}) how a polynomial chaos representation of databases can be constructed. The method is robust and remains efficient for high dimension and large datasets. Based on such a methodology and introducing an additional formulation (based on nonparametric statistics) for the computation of conditional mathematical expectations, we present a novel methodology (Ghanem & Soize, IJNME 2017) for constructing the solution of probabilistic nonconvex constrained optimization problems under uncertainties, using only a fixed small number of function evaluations and probabilistic learning. Such a methodology brings together novel ideas to tackle an outstanding challenge in nonconvex optimization under uncertainties. Several examples are presented to highlight different aspects of the proposed probabilistic learning methodology, in particular for analyzing geophysics, and for complex flows in CFD.