Data-driven polynomial chaos expansion for machine learning regression

Torre, Emiliano; Marelli, Stefano; Embrechts, Paul; Sudret, Bruno

doi:10.1016/j.jcp.2019.03.039

Cited by 132 publications

(85 citation statements)

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“…The generic PCE metamodel for each of the defined model outputs was calculated using the experimental design with a sample size of 2000. The correlation between the model inputs was not considered for building the metamodel to achieve better metamodeling performance [76]. Afterwards, the constructed PCEs were evaluated on the new sample set of size 1,000,000 using coefficients calculated in the previous step.…”

Section: Step C Uncertainty Propagationmentioning

confidence: 99%

Metamodeling for Uncertainty Quantification of a Flood Wave Model for Concrete Dam Breaks

et al. 2020

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Uncertainties in instantaneous dam-break floods are difficult to assess with standard methods (e.g., Monte Carlo simulation) because of the lack of historical observations and high computational costs of the numerical models. In this study, polynomial chaos expansion (PCE) was applied to a dam-break flood model reflecting the population of large concrete dams in Switzerland. The flood model was approximated with a metamodel and uncertainty in the inputs was propagated to the flow quantities downstream of the dam. The study demonstrates that the application of metamodeling for uncertainty quantification in dam-break studies allows for reduced computational costs compared to standard methods. Finally, Sobol’ sensitivity indices indicate that reservoir volume, length of the valley, and surface roughness contributed most to the variability of the outputs. The proposed methodology, when applied to similar studies in flood risk assessment, allows for more generalized risk quantification than conventional approaches.

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Section: Step C Uncertainty Propagationmentioning

confidence: 99%

Metamodeling for Uncertainty Quantification of a Flood Wave Model for Concrete Dam Breaks

et al. 2020

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“…In this section, the proposed ranking based sparse PCE is evaluated on four test cases. The first test case is the Ishigami function [71], the second test case is a ten-dimensional Ackley function, the third case is a waterflooding problem with uncertain permeability field and the forth test case utilizes a data-set from simulations of CO2 injection [72]. In all test cases, the proposed PCE approach is compared to two standard techniques for sparse regression-based PCE: Least Angular Regression [73] and the Orthogonal Matching Pursuit (OMP) algorithm [74].…”

Section: Numerical Examplesmentioning

confidence: 99%

Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models

Tarakanov¹,

Elsheikh²

2019

Journal of Computational Physics

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Surrogate-modelling techniques including Polynomial Chaos Expansion (PCE) is commonly used for statistical estimation (aka. Uncertainty Quantification) of quantities of interests obtained from expensive computational models. PCE is a data-driven regression-based technique that relies on spectral polynomials as basis-functions. In this technique, the outputs of few numerical simulations are used to estimate the PCE coefficients within a regression framework combined with regularization techniques where the regularization parameters are estimated using standard cross-validation as applied in supervised machine learning methods.In the present work, we introduce an efficient method for estimating the PCE coefficients combining Elastic Net regularization with a data-driven feature ranking approach. Our goal is to increase the probability of identifying the most significant PCE components by assigning each of the PCE coefficients a numerical value reflecting the magnitude of the coefficient and its stability with respect to perturbations in the input data. In our evaluations, the proposed approach has shown high convergence rate for high-dimensional problems, where standard feature ranking might be challenging due to the curse of dimensionality.The presented method is implemented within a standard machine learning library (scikit-learn [1]) allowing for easy experimentation with various solvers and regularization techniques (e.g. Tikhonov, LASSO, LARS, Elastic Net) and enabling automatic cross-validation techniques using a widely used and well tested implementation. We present a set of numerical tests on standard analytical functions, a two-phase subsurface flow model and a simulation dataset for CO2 sequestration in a saline aquifer. For all test cases, the proposed approach resulted in a significant increase in PCE convergence rates.

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“…(53) using the ordinary least squares method (Berveiller et al, 2006). To calculate the PCE coefficients of the final, highaccuracy, surrogate M(g(x; w)), the distributions of the input variables are fitted using kernel-smoothing, while retaining the independence assumption, motivated by the results in Torre et al (2018). In addition, a sparse solution is obtained by solving the optimisation problem in Eq.…”

Section: Polynomial Chaos Expansionsmentioning

confidence: 99%

Extending Classical Surrogate Modeling to High Dimensions Through Supervised Dimensionality Reduction: A Data-Driven Approach

Lataniotis

Marelli

Sudret

2020

Int. J. UncertaintyQuantification

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Thanks to their versatility, ease of deployment and high-performance, surrogate models have become staple tools in the arsenal of uncertainty quantification (UQ). From local interpolants to global spectral decompositions, surrogates are characterised by their ability to efficiently emulate complex computational models based on a small set of model runs used for training. An inherent limitation of many surrogate models is their susceptibility to the curse of dimensionality, which traditionally limits their applicability to a maximum of O(10 2 ) input dimensions. We present a novel approach at high-dimensional surrogate modelling that is model-, dimensionality reduction-and surrogate model-agnostic (black box), and can enable the solution of high dimensional (i.e. up to O(10 4 )) problems. After introducing the general algorithm, we demonstrate its performance by combining Kriging and polynomial chaos expansions surrogates and kernel principal component analysis. In particular, we compare the generalisation performance that the resulting surrogates achieve to the classical sequential application of dimensionality reduction followed by surrogate modelling on several benchmark applications, comprising an analytical function and two engineering applications of increasing dimensionality and complexity.

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Data-driven polynomial chaos expansion for machine learning regression

Cited by 132 publications

References 50 publications

Metamodeling for Uncertainty Quantification of a Flood Wave Model for Concrete Dam Breaks

Metamodeling for Uncertainty Quantification of a Flood Wave Model for Concrete Dam Breaks

Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models

Extending Classical Surrogate Modeling to High Dimensions Through Supervised Dimensionality Reduction: A Data-Driven Approach

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