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

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 an high-dimensional input space but with a 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 gradients 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.

A multifidelity approach coupling parameter space reduction and nonintrusive POD with application to structural optimization of passenger ship hulls

“…The novelty of this work is the incorporation into the proper orthogonal decomposition framework of parameter space reduction 22 by constructing a multi‐fidelity surrogate model, 23 , 24 without the need of running simplified simulations. This is done by exploiting the presence of an active subspace 25 of the parameter to reduced state variables map.…”

ATHENA: Advanced Techniques for High dimensional parameter spaces to Enhance Numerical Analysis

“…The reduction affects the dimension of the population of the genetic algorithm at each step. The same interface for Active Subspaces is used in [7] to improve the approximation of Gaussian process regression of scalar functions with low intrinsic dimensionality. The information regarding the presence of an Active Subspace is delivered to a nonlinear multi-fidelity model thus able to reach a higher accuracy.…”

Hull shape design optimization with parameter space and model reductions, and self-learning mesh morphing

“…AS has been also coupled with reduced order methods such as POD-Galerkin [45] in cardiovascular studies, and POD with interpolation [8] and dynamic mode decomposition [49] for CFD applications. Application to multi-fidelity approximations of scalar functions are also presented in [35,27].…”