Bayesian learning of orthogonal embeddings for multi-fidelity Gaussian Processes
This work addresses the challenge of dimensionality reduction in multi-fidelity modeling for engineering applications, but it is incremental as it builds on existing GP and Bayesian methods with specific constraints.
The authors tackled the problem of learning optimal orthogonal embeddings for multi-fidelity Gaussian Processes by developing a Bayesian inference scheme with MCMC sampling, and validated it on synthetic problems and a high-dimensional aerodynamic optimization case, showing improved computational efficiency in handling an 85-dimensional parameterization.
We present a Bayesian approach to identify optimal transformations that map model input points to low dimensional latent variables. The "projection" mapping consists of an orthonormal matrix that is considered a priori unknown and needs to be inferred jointly with the GP parameters, conditioned on the available training data. The proposed Bayesian inference scheme relies on a two-step iterative algorithm that samples from the marginal posteriors of the GP parameters and the projection matrix respectively, both using Markov Chain Monte Carlo (MCMC) sampling. In order to take into account the orthogonality constraints imposed on the orthonormal projection matrix, a Geodesic Monte Carlo sampling algorithm is employed, that is suitable for exploiting probability measures on manifolds. We extend the proposed framework to multi-fidelity models using GPs including the scenarios of training multiple outputs together. We validate our framework on three synthetic problems with a known lower-dimensional subspace. The benefits of our proposed framework, are illustrated on the computationally challenging three-dimensional aerodynamic optimization of a last-stage blade for an industrial gas turbine, where we study the effect of an 85-dimensional airfoil shape parameterization on two output quantities of interest, specifically on the aerodynamic efficiency and the degree of reaction.