Gradient-enhanced kriging

Gradient-enhanced kriging (GEK) is a surrogate modeling technique used in engineering.

A surrogate model (alternatively known as a metamodel, response surface or emulator) is a prediction of the output of an expensive computer code.

[1] This prediction is based on a small number of evaluations of the expensive computer code.

Originally developed for optimization, adjoint solvers are now finding more and more use in uncertainty quantification.

An adjoint solver allows one to compute the gradient of the quantity of interest with respect to all design parameters at the cost of one additional solve.

Now assume that each partial derivative provides as much information for our surrogate as a single primal solve.

Then, the total cost of getting the same amount of information from primal solves only is

[6] When using GEK one takes the following steps: Once the surrogate has been constructed it can be used in different ways, for example for surrogate-based uncertainty quantification (UQ) or optimization.

In a Bayesian framework, we use Bayes' Theorem to predict the Kriging mean and covariance conditional on the observations.

When using GEK, the observations are usually the results of a number of computer simulations.

GEK can be interpreted as a form of Gaussian process regression.

The first method, indirect GEK, defines a small but finite stepsize

, and uses the gradient information to append synthetic data to the observations

Direct GEK is a form of co-Kriging, where we add the gradient information as co-variables.

When we construct direct GEK through the prior covariance matrix, we append the partial derivatives to

[6] [8] Another way of arriving at the same direct GEK predictor is to append the partial derivatives to the observations

[11] Current gradient-enhanced kriging methods do not scale well with the number of sampling points due to the rapid growth in the size of the correlation matrix, where new information is added for each sampling point in each direction of the design space.

To address this issue, a new gradient-enhanced surrogate model approach that drastically reduced the number of hyperparameters through the use of the partial-least squares method that maintains accuracy is developed.

In addition, this method is able to control the size of the correlation matrix by adding only relevant points defined through the information provided by the partial-least squares method.

[12] This approach is implemented into the Surrogate Modeling Toolbox (SMT) in Python (https://github.com/SMTorg/SMT), and it runs on Linux, macOS, and Windows.

A universal augmented framework is proposed in [9] to append derivatives of any order to the observations.

This method can be viewed as a generalization of Direct GEK that takes into account higher-order derivatives.

Also, the observations and derivatives are not required to be measured at the same location under this framework.

[3] The airfoil is operating at a Mach number of 0.8 and an angle of attack of 1.25 degrees.

On the right we see the reference results for the drag coefficient of the airfoil, based on a large number of CFD simulations.

Note that the lowest drag, which corresponds to 'optimal' performance, is close to the undeformed 'baseline' design of the airfoil at (0,0).

In the last figure, we have improved the accuracy of this surrogate model by including the adjoint-based gradient information, indicated by the arrows, and applying GEK.

Example of one-dimensional data interpolated by Kriging and GEK. The black line indicates the test-function, while the gray circles indicate 'observations', 'samples' or 'evaluations' of the test-function. The blue line is the Kriging mean, the shaded blue area illustrates the Kriging standard deviation. With GEK we can add the gradient information, illustrated in red, which increases the accuracy of the prediction.
Transonic airfoil.
Reference results for the drag coefficient of a transonic airfoil, based on a large number of CFD simulations. The horizontal and vertical axis show the deformation of the shape of the airfoil.
Kriging surrogate model of the drag coefficient of a transonic airfoil. The gray dots indicate the configurations for which the CFD solver was run.
GEK surrogate model of the drag coefficient of a transonic airfoil. The gray dots indicate the configurations for which the CFD solver was run, the arrows indicate the gradients.