High-dimension probabilistic learning inference constrained by a stochastic computational model and by target statistical moments in the framework of a small training dataset - Université Gustave Eiffel Accéder directement au contenu
Communication Dans Un Congrès Année : 2023

High-dimension probabilistic learning inference constrained by a stochastic computational model and by target statistical moments in the framework of a small training dataset

Résumé

We present a probabilistic learning inference that allows for integrating data (target set) into a parameterized large stochastic computational model (SCM) resulting from the discretization of a stochastic boundary value problem (BVP). The BVP depends on an uncontrolled random parameter that is in high-dimension (for instance a non-Gaussian random field whose prior probability model is given) and a controlled vector-valued random parameter for which a prior probability model is also given. The target set is relative to a vector-valued random quantity of interest (QoI) defined as an observation of the stochastic solution of the SCM. The probabilistic inference consists in estimating the posterior probability model constrained by the equations of the SCM and by the target set made up of statistical moments of the QoI. A statistical inverse problem has thus to be solved. It is assumed that the evaluation of a single realization of the SCM is numerically expensive. This means that the stochastic solution, and consequently, the random QoI, can only be computed for a small number of realizations of the random controlled and uncontrolled parameters. The training dataset in thus constituted of a small number of points and consequently, a probabilistic learning under constraints must be used for this supervised case for which only a small training dataset is available. The constraints are constituted (i) of statistical moments of the quantity of interest (QoI) for which the targets are given and (ii) of the vector-valued random residue of the stochastic equations of the SCM, random residue that will be minimized in the mean-square sense during the learning process. Therefore, the difficulties of this statistical inverse problem are due to a high numerical cost of the evaluation of a large SCM, to the high stochastic dimension of the random parameters, to the constraints that are defined by an implicit non-injective nonlinear mapping defined on a set in high dimension, and finally, to a small training dataset that is a limitation in machine learning. We propose a novel probabilistic methodology, which makes it possible to circumvent each of these difficulties. The formulation and a few mathematical results will be presented and illustrated through an application in continuum mechanics. This application is a contribution to the three-dimensional stochastic homogenization of heterogeneous linear elastic media in the case of a non-separation of the mesoscale with the macroscale. Consequently, the macroscale is another mesoscale at larger scale with random effective/apparent elastic properties. For the construction of the posterior probability measure using the proposed probabilistic learning inference, the constraints are defined by a target set of given statistical moments of the random effective/apparent elasticity tensor and by the second-order moment of the random normalized residue of the stochastic equations of the SCM. This constraint guarantees that the algorithm seeks to bring the statistical moments closer to their targets while preserving a small residue of the SCM.
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Dates et versions

hal-04146423 , version 1 (30-06-2023)

Identifiants

  • HAL Id : hal-04146423 , version 1

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Christian Soize. High-dimension probabilistic learning inference constrained by a stochastic computational model and by target statistical moments in the framework of a small training dataset. (General lecture), 14th International Symposium on Continuum Models and Discrete Systems, CMDS 14, Jun 2023, Paris, France. ⟨hal-04146423⟩
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