Probabilistic learning inference of boundary value problem with uncertainties based on Kullback–Leibler divergence under implicit constraints

Soize, Christian

doi:10.1016/j.cma.2022.115078

Cited by 7 publications

(6 citation statements)

References 102 publications

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“…The second part of this paper corresponds to Section 6 in which we present the methodology to construct the posterior probability measure based on the use of the Kullback-Leiber minimum principle with the prior model and the target set. This methodology is similar to the one we have used in [38,51], but for which the constraints are now the one presented in Section 5. Thus the mathematical proofs are adapted and modified because the hypotheses are no longer the same.…”

Section: Organization Of the Papermentioning

confidence: 99%

“…In this section we reuse part of the developments that we presented in paper [51]. We do not want to limit ourselves to referring the reader to this reference, because the hypotheses are not the same, the Lemmas and Theorems must be reformulated, and their proofs must be adapted and modified.…”

Section: Kullback-leibler Minimum Principle For Estimating the Poster...mentioning

confidence: 99%

“…. , H post,ν ) is estimated by using the Kullback-Leibler divergence minimum principle [30,31,32,38,51]. This estimation of p post H is performed by using the prior pdf η → p H (η) on R ν in which p H is defined by Eqs.…”

Section: Posterior Estimate Using the Kullback-leibler Divergence Min...mentioning

confidence: 99%

“…Presently we propose to use the Störmer-Verlet scheme (see [72] for the deterministic case and [73] for the stochastic case), which is an efficient scheme that allows for having a long-time energy conservation for non-dissipative Hamiltonian dynamical systems. In [74], we have proposed to use an extension of the Störmer-Verlet scheme for stochastic dissipative Hamiltonian systems, that we have also used in [75,59,40,38,39,47,51].…”

Section: Numerical Implementationmentioning

confidence: 99%

“…data) in a supervised model, which is defined only by a small number of points in a training set and for which the target set of realizations are specified only for the quantities of interest (output) and not for the control variable (input). Finally, it should be noted that, when the training set is generated using a stochastic boundary value problem, there is also another method as we have proposed and validated in [39,51]. It consists, for the generation of the constrained learned set, to introduce an additional scalar constraint to minimize the norm of the residue of the partial differential equations of the boundary value problem.…”

Section: Training Setmentioning

confidence: 99%

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Probabilistic learning constrained by realizations using a weak formulation of Fourier transform of probability measures

Soize¹

2022

Preprint

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This paper deals with the taking into account a given set of realizations as constraints in the Kullback-Leibler minimum principle, which is used as a probabilistic learning algorithm. This permits the effective integration of data into predictive models. We consider the probabilistic learning of a random vector that is made up of either a quantity of interest (unsupervised case) or the couple of the quantity of interest and a control parameter (supervised case).A training set of independent realizations of this random vector is assumed to be given and to be generated with a prior probability measure that is unknown. A target set of realizations of the QoI is available for the two considered cases. The framework is the one of non-Gaussian problems in high dimension. A functional approach is developed on the basis of a weak formulation of the Fourier transform of probability measures (characteristic functions). The construction makes it possible to take into account the target set of realizations of the QoI in the Kullback-Leibler minimum principle. The proposed approach allows for estimating the posterior probability measure of the QoI (unsupervised case) or of the posterior joint probability measure of the QoI with the control parameter (supervised case). The existence and the uniqueness of the posterior probability measure is analyzed for the two cases. The numerical aspects are detailed in order to facilitate the implementation of the proposed method. The presented application in high dimension demonstrates the efficiency and the robustness of the proposed algorithm.

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Section: Organization Of the Papermentioning

confidence: 99%

Section: Kullback-leibler Minimum Principle For Estimating the Poster...mentioning

confidence: 99%

Section: Posterior Estimate Using the Kullback-leibler Divergence Min...mentioning

confidence: 99%

Section: Numerical Implementationmentioning

confidence: 99%

Section: Training Setmentioning

confidence: 99%

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Probabilistic learning constrained by realizations using a weak formulation of Fourier transform of probability measures

Soize¹

2022

Preprint

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Probabilistic Learning Inference Constrained by an Uncertain Model and a Target: A General Method with Application to Elasticity Homogenization Without Scale Separation

Soize

2024

Springer Proceedings in Mathematics &Amp; Statistics

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Probabilistic learning constrained by realizations using a weak formulation of Fourier transform of probability measures

Soize

2022

Comput Stat

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Probabilistic learning inference of boundary value problem with uncertainties based on Kullback–Leibler divergence under implicit constraints

Cited by 7 publications

References 102 publications

Probabilistic learning constrained by realizations using a weak formulation of Fourier transform of probability measures

Probabilistic learning constrained by realizations using a weak formulation of Fourier transform of probability measures

Probabilistic Learning Inference Constrained by an Uncertain Model and a Target: A General Method with Application to Elasticity Homogenization Without Scale Separation

Probabilistic learning constrained by realizations using a weak formulation of Fourier transform of probability measures

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