Bayesian learning of orthogonal embeddings for multi-fidelity Gaussian Processes

TL;DR

This paper introduces a Bayesian method for learning orthogonal embeddings in Gaussian Processes, enabling efficient dimensionality reduction and multi-fidelity modeling, validated on synthetic and real-world aerodynamic optimization problems.

Contribution

It develops a Bayesian inference framework with Geodesic Monte Carlo for orthogonal projections in GPs, extending to multi-fidelity models with joint output training.

Findings

Effective identification of low-dimensional subspaces in synthetic tests.

Improved aerodynamic optimization results with high-dimensional input parameters.

Demonstrated scalability to complex, real-world engineering problems.

Abstract

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.