Multi-fidelity data fusion for the approximation of scalar functions with low intrinsic dimensionality through active subspaces

TL;DR

This paper introduces a multi-fidelity Gaussian process regression method that leverages active subspaces to efficiently approximate high-dimensional scalar functions with low intrinsic dimensionality, especially useful when data is scarce.

Contribution

The work extends multi-fidelity Gaussian process models by integrating active subspaces to handle high-dimensional inputs with low intrinsic dimensionality, reducing the need for extensive simulations.

Findings

Effective in high-dimensional benchmarks

Improves accuracy with limited data

Utilizes active subspaces for dimensionality reduction

Abstract

Gaussian processes are employed for non-parametric regression in a Bayesian setting. They generalize linear regression, embedding the inputs in a latent manifold inside an infinite-dimensional reproducing kernel Hilbert space. We can augment the inputs with the observations of low-fidelity models in order to learn a more expressive latent manifold and thus increment the model's accuracy. This can be realized recursively with a chain of Gaussian processes with incrementally higher fidelity. We would like to extend these multi-fidelity model realizations to case studies affected by a high-dimensional input space but with low intrinsic dimensionality. In this cases physical supported or purely numerical low-order models are still affected by the curse of dimensionality when queried for responses. When the model's gradient information is provided, the presence of an active subspace can be exploited to design low-fidelity response surfaces and thus enable Gaussian process multi-fidelity regression, without the need to perform new simulations. This is particularly useful in the case of data scarcity. In this work we present a multi-fidelity approach involving active subspaces and we test it on two different high-dimensional benchmarks.