Bayesian Surrogate Analysis and Uncertainty Propagation

TL;DR

This paper introduces a Bayesian framework for quantifying and propagating surrogate model uncertainties in computational simulations, improving the reliability of uncertainty estimates especially when surrogate trustworthiness is questionable.

Contribution

It develops a Bayesian approach that naturally incorporates surrogate uncertainties into simulation output uncertainties, extending existing methods like Polynomial Chaos and Gaussian Process Regression.

Findings

Bayesian surrogate analysis provides a natural trust measure.

Surrogate uncertainties can be integrated into output uncertainty estimates.

The framework is applicable to various surrogate models and includes spatio-temporal correlations.

Abstract

The quantification of uncertainties of computer simulations due to input parameter uncertainties is paramount to assess a model's credibility. For computationally expensive simulations, this is often feasible only via surrogate models that are learned from a small set of simulation samples. The surrogate models are commonly chosen and deemed trustworthy based on heuristic measures, and substituted for the simulation in order to approximately propagate the simulation input uncertainties to the simulation output. In the process, the contribution of the uncertainties of the surrogate itself to the simulation output uncertainties are usually neglected. In this work, we specifically address the case of doubtful surrogate trustworthiness, i.e. non-negligible surrogate uncertainties. We find that Bayesian probability theory yields a natural measure of surrogate trustworthiness, and that surrogate uncertainties can easily be included in simulation output uncertainties. For a Gaussian likelihood for the simulation data, with unknown surrogate variance and given a generalized linear surrogate model, the resulting formulas reduce to simple matrix multiplications. The framework contains Polynomial Chaos Expansions as a special case, and is easily extended to Gaussian Process Regression. Additionally, we show a simple way to implicitly include spatio-temporal correlations. Lastly, we demonstrate a numerical example where surrogate uncertainties are in part negligible and in part non-negligible.