Worst-Case Learning under a Multi-fidelity Model

TL;DR

This paper develops a worst-case guarantee framework for multi-fidelity surrogate modeling, providing deterministic error bounds for high-fidelity code approximations using combined low- and high-fidelity data.

Contribution

It introduces a novel theoretical framework based on Optimal Recovery for designing multi-fidelity surrogates with deterministic error guarantees, extending to new scenarios in Hilbert spaces.

Findings

New theoretical results for optimal estimation of linear functionals

Optimal approximation of quantities of interest in Hilbert spaces

Determination of Chebyshev centers for hyperellipsoid intersections

Abstract

Inspired by multi-fidelity methods in computer simulations, this article introduces procedures to design surrogates for the input/output relationship of a high-fidelity code. These surrogates should be learned from runs of both the high-fidelity and low-fidelity codes and be accompanied by error guarantees that are deterministic rather than stochastic. For this purpose, the article advocates a framework tied to a theory focusing on worst-case guarantees, namely Optimal Recovery. The multi-fidelity considerations triggered new theoretical results in three scenarios: the globally optimal estimation of linear functionals, the globally optimal approximation of arbitrary quantities of interest in Hilbert spaces, and their locally optimal approximation, still within Hilbert spaces. The latter scenario boils down to the determination of the Chebyshev center for the intersection of two hyperellipsoids. It is worth noting that the mathematical framework presented here, together with its possible extension, seems to be relevant in several other contexts briefly discussed.