Erratum: Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces

Constantine, Paul G.; Dow, Eric; Wang, Qiqi

doi:10.1137/140983598

Cited by 5 publications

(4 citation statements)

References 1 publication

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“…Both approaches share the fundamental requirement for an appropriate adhoc parametrisation of the PDE system. In addition to being cumbersome, such parametrisation are, in the case of meta-models based on polynomial chaos expansion or Kriging, further restricted to be of low-dimension to circumvent the curse of dimensionality [6].…”

Section: Introductionmentioning

confidence: 99%

A Bayesian multiscale CNN framework to predict local stress fields in structures with microscale features

Krokos

Xuan²,

Bordas

et al. 2021

Comput Mech

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Multiscale computational modelling is challenging due to the high computational cost of direct numerical simulation by finite elements. To address this issue, concurrent multiscale methods use the solution of cheaper macroscale surrogates as boundary conditions to microscale sliding windows. The microscale problems remain a numerically challenging operation both in terms of implementation and cost. In this work we propose to replace the local microscale solution by an Encoder-Decoder Convolutional Neural Network that will generate fine-scale stress corrections to coarse predictions around unresolved microscale features, without prior parametrisation of local microscale problems. We deploy a Bayesian approach providing credible intervals to evaluate the uncertainty of the predictions, which is then used to investigate the merits of a selective learning framework. We will demonstrate the capability of the approach to predict equivalent stress fields in porous structures using linearised and finite strain elasticity theories.

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Section: Introductionmentioning

confidence: 99%

A Bayesian multiscale CNN framework to predict local stress fields in structures with microscale features

Krokos

Xuan²,

Bordas

et al. 2021

Comput Mech

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“…In this section, we discuss the classic approach to discovering the active subspace using gradient information [69,67,68,73,70,74,75,90,91]. Recall that we are dealing with a high-dimensional response surface, and that we would like to approximate it as in Eq.…”

Section: Gradient-based Approach To Active Subspace Regressionmentioning

confidence: 99%

“…Mathematically, an AS is described by an orthogonal matrix that projects the original inputs to this low-dimensional manifold. The classic framework for discovering the AS was laid down by Constantine [67][68][69][70]. One builds a positive-definite matrix that depends upon the gradients of the response surface.…”

Section: Introductionmentioning

confidence: 99%

“…AS applications focus on computer experiments [67][68][69][70], while PLS has been extensively used to model real experiments with high-dimensional inputs/outputs in the field of chemometrics [79][80][81]. PLS not only projects the input to a lower dimensional space using an orthogonal projection matrix, but, if required, it can do the same to a high-dimensional output.…”

Section: Introductionmentioning

confidence: 99%

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Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

Tripathy

Bilionis

Gonzalez

2016

Journal of Computational Physics

164

126

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Uncertainty quantification (UQ) tasks, such as model calibration, uncertainty propagation, and optimization under uncertainty, typically require several thousand evaluations of the underlying computer codes. To cope with the cost of simulations, one replaces the real response surface with a cheap surrogate based, e.g., on polynomial chaos expansions, neural networks, support vector machines, or Gaussian processes (GP). However, the number of simulations required to learn a generic multivariate response grows exponentially as the input dimension increases. This curse of dimensionality can only be addressed, if the response exhibits some special structure that can be discovered and exploited. A wide range of physical responses exhibit a special structure known as an active subspace (AS). An AS is a linear manifold of the stochastic space characterized by maximal response variation. The idea is that one should first identify this low dimensional manifold, project the high-dimensional input onto it, and then link the projection to the output. If the dimensionality of the AS is low enough, then learning the link function is a much easier problem than the original problem of learning a high-dimensional function. The classic approach to discovering the AS requires gradient information, a fact that severely limits its applicability. Furthermore, and partly because of its reliance to gradients, it is not able to handle noisy observations. The latter is an essential trait if one wants to be able to propagate uncertainty through stochastic simulators, e.g., through molecular dynamics codes. In this work, we develop a probabilistic version of AS which is gradient-free and robust to observational noise. Our approach relies on a novel Gaussian process regression with built-in dimensionality reduction. In particular, the AS is represented as an orthogonal projection matrix that serves as yet another covariance function hyper-parameter to be estimated from the data. To train the model, we design a two-step maximum likelihood optimization procedure that ensures the orthogonality of the projection matrix by exploiting recent results on the Stiefel manifold, i.e., the manifold of matrices with orthogonal columns. The additional benefit of our probabilistic formulation, is that it allows us to select the dimensionality of the AS via the Bayesian information criterion. We validate our approach by showing that it can discover the right AS in synthetic examples without gradient information using both noiseless and noisy observations. We demonstrate that our method is able to discover the same AS as the classical approach in a challenging one-hundred-dimensional problem involving an elliptic stochastic partial differential equation with random conductivity. Finally, we use our approach to study the effect of geometric and material uncertainties in the propagation of solitary waves in a one dimensional granular system.

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Machine learning for high-dimensional dynamic stochastic economies

Scheidegger

Bilionis

2019

Journal of Computational Science

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Erratum: Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces

Cited by 5 publications

References 1 publication

A Bayesian multiscale CNN framework to predict local stress fields in structures with microscale features

A Bayesian multiscale CNN framework to predict local stress fields in structures with microscale features

Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

Machine learning for high-dimensional dynamic stochastic economies

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