A survey on multi-fidelity surrogates for simulators with functional outputs: unified framework and benchmark

TL;DR

This study presents a unified framework for multi-fidelity surrogates of simulators producing functional outputs, integrating dimensionality reduction with intermediate latent surrogates. It benchmarks more than a dozen surrogate variants across viscous-ball dynamics and airfoil CP fields, using two fidelity levels and modular design to accommodate different meshes. The results show that most multi-fidelity approaches outperform single-fidelity baselines, but no method universally dominates; performance hinges on fidelity correlation, training data size, and the nature of residuals, with linear DR (PCA) paired with AR1-based fusion or corrective strategies offering robust, practical options. Practitioners are guided to start with linear DR, leverage manifold alignment when necessary, and selectively adopt nonlinear DR or alternative PCA variants based on problem-specific information content and computational budgets.

Abstract

Multi-fidelity surrogate models combining dimensionality reduction and an intermediate surrogate in the reduced space allow a cost-effective emulation of simulators with functional outputs. The surrogate is an input-output mapping learned from a limited number of simulator evaluations. This computational efficiency makes surrogates commonly used for many-query tasks. Diverse methods for building them have been proposed in the literature, but they have only been partially compared. This paper introduces a unified framework encompassing the different surrogate families, followed by a methodological comparison and the exposition of practical considerations. More than a dozen of existing multi-fidelity surrogates have been implemented under the unified framework and evaluated on a set of benchmark problems. Based on the results, guidelines and recommendations are proposed regarding multi-fidelity surrogates with functional outputs. Our study shows that most multi-fidelity surrogates outperform their tested single-fidelity counterparts under the considered settings. But no particular surrogate is performing better on every test case. Therefore, the selection of a surrogate should consider the specific properties of the emulated functions, in particular the correlation between the low- and high-fidelity simulators, the size of the training set, the local nonlinear variations in the residual fields, and the size of the training datasets.

Figures (32)

• Figure 1: Sketch of a generic multi-fidelity simulation, with the input vector $\mathbf{u}\in U$ and at fidelity level $l\in\{1,\dots,L\}$ the simulator $\mathcal{S}_l$, the mesh $\mathbf{X}_l\in X_l$ and the scalar output field $\mathbf{y}_l\in Y_l$.
• Figure 2: Diagram of the corrective approaches malouinInterpolation2013
• Figure 3: Diagram of the mapping approach, toalPotential2014 in red, wangMultifidelity2020 in yellow. The arrows represent the input and output of the single-fidelity intermediate surrogate $\mathcal{IS}$.
• Figure 4: Diagram of the fusion approaches, rokitaMultifidelity2018benamaraMultifidelity2017mifsudVariablefidelity2016thenonMultifidelity2016 in red, kerleguerMultifidelity2023 in yellow, bunnellMultifidelity2021 in blue, perronMultifidelity2021deckerManifold2022 in pink
• Figure 5: Simulation setup for the free fall of a ball in a fluid medium case.