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

Abstract

Multi-fidelity models are of great importance due to their capability of fusing information coming from different simulations and sensors. Gaussian processes are employed for nonparametric 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 these 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 existence of an active subspace, or a nonlinear transformation of the input parameter space, 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 new multi-fidelity approach involving active subspaces and nonlinear level-set learning method. We test the proposed numerical method on two different high-dimensional benchmarks, and on a more complex car aerodynamics problem. We show how a low intrinsic dimensionality bias can increase the accuracy of GP response surfaces.

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

“…Future works will focus on improving the accuracy of constraints evaluations, for example with a multi-fidelity approximation of the scalar output and not only for the reconstruction of the entire field. 23 Another possibility is the exploitation of local information with local active subspaces 64 or nonlinear techniques, based on kernels 65 or level-sets, 66,67 to further improve the regression performance of the low-fidelity model. Other physical constraints can also be considered such as the position of the center of mass.…”

“…Moreover we exploit active subspaces 25 (AS) for the reduction of the parameter space dimension to build low-fidelity models and improve the PODI prediction capabilities in a multi-fidelity setting 24,36 called nonlinear autoregressive multi-fidelity Gaussian process regression with active subspaces (NARGPAS). 23 These parameter and model reduction methods are combined for a computational efficient and reliable evaluation of the constraints regarding the stability of the whole hull: we check how many elements are yielded, and how many elements are subjected to buckling phenomena. We remark that we allow local stress peaks to exceed the classification society rule limits, since we automatically incorporate within the function to optimize the necessary interventions at the shipyard to stabilize such elements.…”

“…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.…”

Kernel‐based active subspaces with application to computational fluid dynamics parametric problems using the discontinuous Galerkin method

“…Comparison between the sufficiency summary plots obtained from the application of AS and KAS methods for the surface of revolution model function with domain [−3, 3] 2 , defined in Equation (26). The left plot refers to AS, the right plot to KAS.…”

Parameter space and model order reduction for the structural optimisation of passenger ships

“…We exploit proper orthogonal decomposition (POD) to reduce the dimensionality of the stress tensor field and the active subspaces technique for the parameter space. We propose a multi-fidelity approach, where the low-fidelity model is built through parameter space reduction without the need of running any simplified simulation [2]. We integrate a low-dimensionality bias within a nonlinear autoregressive scheme of Gaussian processes.…”