An Overview of Gradient-Enhanced Metamodels with Applications

  • Luc LaurentEmail author
  • Rodolphe Le Riche
  • Bruno Soulier
  • Pierre-Alain Boucard
Original Paper


Metamodeling, the science of modeling functions observed at a finite number of points, benefits from all auxiliary information it can account for. Function gradients are a common auxiliary information and are useful for predicting functions with locally changing behaviors. This article is a review of the main metamodels that use function gradients in addition to function values. The goal of the article is to give the reader both an overview of the principles involved in gradient-enhanced metamodels while also providing insightful formulations. The following metamodels have gradient-enhanced versions in the literature and are reviewed here: classical, weighted and moving least squares, Shepard weighting functions, and the kernel-based methods that are radial basis functions, kriging and support vector machines. The methods are set in a common framework of linear combinations between a priori chosen functions and coefficients that depend on the observations. The characteristics common to all kernel-based approaches are underlined. A new \(\nu \)-GSVR metamodel which uses gradients is given. Numerical comparisons of the metamodels are carried out for approximating analytical test functions. The experiments are replicable, as they are performed with an opensource available toolbox. The results indicate that there is a trade-off between the better computing time of least squares methods and the larger versatility of kernel-based approaches.


\(\nu \)-GSVR

\(\nu \)-Version of the gradient-enhanced support vector regression (surrogate model)

\(\varepsilon \)-SVR

\(\varepsilon \)-Version of the support vector regression (surrogate model)

\(\nu \)-SVR

\(\nu \)-Version of the support vector regression (surrogate model)

\(\varepsilon _k\)-GSVR

\(\varepsilon _k\)-Version of the gradient-enhanced support vector regression (surrogate model)


Best linear unbiased predictor


Efficient global optimization [33]


Gradient-based kriging (surrogate model)


Gradient-enhanced kriging (surrogate model)


Gradient-enhanced universal kriging (surrogate model)


Gradient-enhanced cokriging (surrogate model, same as GBK, GEK and GEUK)


Generalized least square regression (surrogate model)


Gradient-enhanced least square regression (surrogate model)


Gradient-enhanced radial basis function (surrogate model)


Gradient-enhanced approximation toolbox (Matlab/Octave’s toolbox [12])


Gradient-enhanced support vector regression (surrogate model)


Inverse distance weighting method also called Shepard weighting method (surrogate model)


Improved hypercube sampling (sampling technique [103])


Indirect gradient-enhanced ordinary kriging (surrogate model)


Indirect gradient-enhanced radial basis function (surrogate model)


Kriging (surrogate model)




Least square regression (surrogate model)


Moving least square regression (surrogate model)


Mean square error (quality criterion)


Multiple design of experiments (Matlab/Octave’s toolbox [133])


Ordinary kriging (surrogate model)


Gradient-enhanced ordinary cokriging (surrogate model)


Radial basis function (surrogate model)


Response surface methodology


Surrogate-based analysis and optimization [18]


Support vector machine


Support vector regression (surrogate model)


Weigthed least square regression (surrogate model)


Multivariate adaptive regression splines (surrogate model)


Artificial neural network (surrogate model)


Latin hypercube sampling (sampling technique)


Orthogonal array (sampling technique)


Uniform design (sampling technique)


Relative average absolute error (quality criterion)


Root mean square error (quality criterion)


Relative maximal absolute error (quality criterion)


Leave-one-out cross-validation


Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© CIMNE, Barcelona, Spain 2017

Authors and Affiliations

  1. 1.Laboratoire de Mécanique des Structures et des Systèmes CouplésConservatoire National des Arts et MétiersParisFrance
  2. 2.École Nationale Supérieure des Mines de Saint-ÉtienneSaint-ÉtienneFrance
  3. 3.LMT Cachan (ENS Cachan/CNRS/Université Paris-Saclay)CachanFrance

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