Adaptive Dimensionality Reduction for Fast Sequential Optimization With Gaussian Processes

Abstract

Available computational models for many engineering design applications are both expensive and and of a black-box nature. This renders traditional optimization techniques difficult to apply, including gradient-based optimization and expensive heuristic approaches. For such situations, Bayesian global optimization approaches, that both explore and exploit a true function while building a metamodel of it, are applied. These methods often rely on a set of alternative candidate designs over which a querying policy is designed to search. For even modestly high-dimensional problems, such an alternative set approach can be computationally intractable, due to the reliance on excessive exploration of the design space. To overcome this, we have developed a framework for the optimization of expensive black-box models, which is based on active subspace exploitation and a two-step knowledge gradient policy. We demonstrate our approach on three benchmark problems and a practical aerostructural wing design problem, where our method performs well against traditional direct application of Bayesian global optimization techniques.

Enhancing CFD predictions in shape design problems by model and parameter space reduction

“…Active subspaces [10] have been successfully employed in many engineering fields [12,13]. Among other we mention applications in shape optimization [20,38], combustion simulations [29], and in naval engineering [51]. For multifidelity dimension reduction with AS see [32], for multivariate extension of AS we mention [58], while for a coupling with deep neural networks see [52].…”

“…To speed up the convergence to the regime state (t = 30 s) we applied the DMD to get the future-state prediction of the lift. In particular, due to the initial propagation of the boundary conditions, for all the 70 training deformations we use the trend of lift coefficients within the temporal interval [12,20] s to fit the DMD model, that means 8000 temporal information (Δt = 0.001 s). Since we used 10 POD modes-selected using the energetic criterion-for the projection of the DMD operator, our low-rank operator results of dimension 10.…”

“…Sensitivity analysis of the dimension of the training set for the DMD (left) and for the response surface using GPR (right). For the DMD, we use 70 samples (of the parametric space) evolving in time in[12,20] s and we measure the mean relative error at time 30 s varying the sampling frequency; for the GPR, we build the response surface using up to 70 sampling lift coefficients at time 20 s and computing the mean relative error over the test dataset composed by 100 test deformations reduction of the simulation time required to the FOM as the remaining time is simulated by an approximated model. On the other side, once the reduced order model has been constructed, exploiting the combination of the Gaussian Process approximation and the DMD, it is possible to test new geometries in real time, with a negligible computational cost.…”

Multi-fidelity data fusion through parameter space reduction with applications to automotive engineering

“…Active subspaces (AS) [7,55] can be used to build a surrogate low-fidelity model with reduced input space taking advantage of the correlations of the model's gradients when available. Reduction in parameter space through AS has been proven successful in a diverse range of applications such as: shape optimization [29,15,12,10], car aerodynamics studies [33], hydrologic models [19], naval and nautical engineering [51,31], coupled with intrusive reduced order methods in cardiovascular studies [48], in CFD problems in a data-driven setting [11,50]. A kernel-based extension of AS for both scalar and vectorial functions can be found in [40], while for a new local approach to parameter space reduction see [41].…”

Kernel-based Active Subspaces with application to CFD parametric problems using Discontinuous Galerkin method

“…Advances in efficient global design optimization with surrogate modeling are presented in [35,34] and applied to the shape design of the N + 2 Supersonic Passenger Jet. Applications to enhance optimization methods have been developed in [59,20,17].…”