A new formulation of gradient-enhanced surrogate model and application to aerodynamic design

Gradient-enhanced surrogate modeling (such as gradient-enhanced kriging model) has received much attention in the research area of design optimization, due to the availability of inexpensive gradient computation approaches such as adjoint method or automatic differentiation. Theoretically, if the cheap gradients are employed to build a gradient-enhanced kriging (GEK) model, the efficiency of training a sufficiently accurate surrogate model for a high-dimensional unknown function can be greatly improved. However, in fact, the building of a GEK model is suffering from the large computational cost associated with inverting the “large” correlation matrix of order of (m+1)n, where m is the numbers of independent (or design) variables and n is the number of sampling sites. This situation becomes extremely serious when m is large. This paper proposes a new formulation of GEK model to work out this problem and applies it to aerodynamic design. The core idea is to build a series of GEK sub-models with much smaller correlation matrix, then to sum them up with appropriate weight coefficients. We make a self-contained derivation about the mathematical formulation of the new GEK predictor and its mean-squared error, and present the method of tuning the hyper-parameters of the model. The new GEK model is verified by analytical problems as well as surrogate modeling for drag coefficient of an RAE2822 airfoil, with the adjoint method computing the gradients. It is shown, by using the proposed GEK, the efficiency of building the surrogate model is significantly improved, with a price paid for the slightly decreased accuracy when compared to the conventional GEK model. The effectiveness of the new GEK model is also demonstrated for inverse design of transonic wings with the number of design variables in the range from 36 to 54, which shows that the new GEK model is more efficient for the problems with higher dimensional design space.