Entropy-based closure for probabilistic learning on manifolds

Soize, Christian; Ghanem, Roger; Safta, Cosmin; Huan, Xun; Vane, Zachary P.; Oefelein, Joseph C.; Lacaze, Guilhem; Najm, Habib N.; Tang, Qi; Chen, X.

doi:10.1016/j.jcp.2018.12.029

Cited by 25 publications

(23 citation statements)

References 52 publications

(109 reference statements)

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“…The construction introduces two hyperparameters: the dimension m ≤ N and the smoothing parameter ε diff >0. An algorithm is proposed in the work of Soize et al for estimating their values. Most of the time, m and ε diff can be chosen as follows.…”

Section: Computational Statistical Methods For Generating Realizationsmentioning

confidence: 99%

“…If function

\overset{m}{true}

is a decreasing function of ε diff in the broad sense (if not, see the work of Soize et al), then the optimal value

ε_{diff}^{opt}

of ε diff can be chosen as the smallest value of the integer

\overset{m}{true} false(ε_{diff}^{opt} false)

such that

({], \overset{m}{true} ((), ε_{diff}^{opt}) < \overset{m}{true} false(ε_{diff} false), \forall ε_{diff} \in) 0, ε_{diff}^{opt} false[false} \cap ({], \overset{m}{true} ((), ε_{diff}^{opt}) = \overset{m}{true} false(ε_{diff} false), \forall ε_{diff} \in) ε_{diff}^{opt}, 1.5 ε_{diff}^{opt} false[false} .

…”

Section: Computational Statistical Methods For Generating Realizationsmentioning

confidence: 99%

“…If functionm is a decreasing function of diff in the broad sense (if not, see the work of Soize et al 40 ), then the optimal value opt diff of diff can be chosen as the smallest value of the integerm(…”

Section: Construction Of the Diffusion-maps Basismentioning

confidence: 99%

“…This paper is devoted to this framework: learning from a high‐dimensional small data set, without invoking the Gaussian assumption. The first ingredient of the proposed approach is the probabilistic learning on manifolds (PLoM) presented in the work of Soize and Ghanem, for which complementary developments and applications with validation are presented in other works. This PLoM allows for constructing a generated data set

D_{M}^{g}

made up of M additional independent realizations

false{{boldx}_{ar}^{ℓ}, ℓ = 1, …, M false}

of a non‐Gaussian random vector X , defined on a probability space

false(normalΘ, scriptT, scriptP false)

, with values in

{double-struckR}^{n}

, for which the probability distribution P X ( d x ) is unknown, using only an initial data set D N (training set), which is a small data set, made up of N independent realizations { x j , j =1,…, N } of X with x j = X ( θ j ) for θ j ∈Θ and such that M ≫ N .…”

Section: Introductionmentioning

confidence: 99%

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Physics‐constrained non‐Gaussian probabilistic learning on manifolds

Soize

Ghanem

2019

Numerical Meth Engineering

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Summary An extension of the probabilistic learning on manifolds (PLoM), recently introduced by the authors, has been presented: In addition to the initial data set given for performing the probabilistic learning, constraints are given, which correspond to statistics of experiments or of physical models. We consider a non‐Gaussian random vector whose unknown probability distribution has to satisfy constraints. The method consists in constructing a generator using the PLoM and the classical Kullback‐Leibler minimum cross‐entropy principle. The resulting optimization problem is reformulated using Lagrange multipliers associated with the constraints. The optimal solution of the Lagrange multipliers is computed using an efficient iterative algorithm. At each iteration, the Markov chain Monte Carlo algorithm developed for the PLoM is used, consisting in solving an Itô stochastic differential equation that is projected on a diffusion‐maps basis. The method and the algorithm are efficient and allow the construction of probabilistic models for high‐dimensional problems from small initial data sets and for which an arbitrary number of constraints are specified. The first application is sufficiently simple in order to be easily reproduced. The second one is relative to a stochastic elliptic boundary value problem in high dimension.

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Section: Computational Statistical Methods For Generating Realizationsmentioning

confidence: 99%

“…If function

\overset{m}{true}

is a decreasing function of ε diff in the broad sense (if not, see the work of Soize et al), then the optimal value

ε_{diff}^{opt}

of ε diff can be chosen as the smallest value of the integer

\overset{m}{true} false(ε_{diff}^{opt} false)

such that

({], \overset{m}{true} ((), ε_{diff}^{opt}) < \overset{m}{true} false(ε_{diff} false), \forall ε_{diff} \in) 0, ε_{diff}^{opt} false[false} \cap ({], \overset{m}{true} ((), ε_{diff}^{opt}) = \overset{m}{true} false(ε_{diff} false), \forall ε_{diff} \in) ε_{diff}^{opt}, 1.5 ε_{diff}^{opt} false[false} .

…”

Section: Computational Statistical Methods For Generating Realizationsmentioning

confidence: 99%

Section: Construction Of the Diffusion-maps Basismentioning

confidence: 99%

D_{M}^{g}

made up of M additional independent realizations

false{{boldx}_{ar}^{ℓ}, ℓ = 1, …, M false}

of a non‐Gaussian random vector X , defined on a probability space

false(normalΘ, scriptT, scriptP false)

, with values in

{double-struckR}^{n}

Section: Introductionmentioning

confidence: 99%

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Physics‐constrained non‐Gaussian probabilistic learning on manifolds

Soize

Ghanem

2019

Numerical Meth Engineering

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“…Clearly, these eigenvalues and hence the optimal value of m depend on , the width of the diffusion kernel. A maximum entropy argument [29] is used to simultaneously select the values of and m.…”

Section: Diffusion Mapsmentioning

confidence: 99%

Design optimization of a scramjet under uncertainty using probabilistic learning on manifolds

Ghanem

Soize²,

Safta

et al. 2019

Journal of Computational Physics

Self Cite

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We demonstrate, on a scramjet combustion problem, a constrained probabilistic learning approach that augments physics-based datasets with realizations that adhere to underlying constraints and scatter. The constraints are captured and delineated through diffusion maps, while the scatter is captured and sampled through a projected stochastic differential equation. The objective function and constraints of the optimization problem are then efficiently framed as non-parametric conditional expectations. Different spatial resolutions of a large-eddy simulation filter are used to explore the robustness of the model to the training dataset and to gain insight into the significance of spatial resolution on optimal design.

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Surrogate modeling of structural seismic response using probabilistic learning on manifolds

Zhong

Navarro

Govindjee

et al. 2023

Earthq Engng Struct Dyn

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Nonlinear response history analysis (NLRHA) is generally considered to be a reliable and robust method to assess the seismic performance of buildings under strong ground motions. While NLRHA is fairly straightforward to evaluate individual structures for a select set of ground motions at a specific building site, it becomes less practical for performing large numbers of analyses to evaluate either (1) multiple models of alternative design realizations with a site-specific set of ground motions, or (2) individual archetype building models at multi-This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

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Entropy-based closure for probabilistic learning on manifolds

Cited by 25 publications

References 52 publications

Physics‐constrained non‐Gaussian probabilistic learning on manifolds

Physics‐constrained non‐Gaussian probabilistic learning on manifolds

Design optimization of a scramjet under uncertainty using probabilistic learning on manifolds

Surrogate modeling of structural seismic response using probabilistic learning on manifolds

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