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A probabilistic learning on manifolds as a new tool in machine learning and data science with applications in computational mechanics

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.
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Contributor : Christian Soize <>
Submitted on : Friday, July 5, 2019 - 6:23:57 PM
Last modification on : Saturday, April 4, 2020 - 11:06:17 AM


  • HAL Id : hal-02175533, version 1



Christian Soize. A probabilistic learning on manifolds as a new tool in machine learning and data science with applications in computational mechanics. (Plenary Lecture), UNCECOMP 2019, 3nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering, and COMPDYN 2019, 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Jun 2019, Island of Crete, Greece. ⟨hal-02175533⟩



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