Machine Learning based optimization for interval uncertainty propagation

TL;DR

This paper introduces two machine learning-based optimization methods using Bayesian optimization and Gaussian Process regression to efficiently estimate response bounds in engineering systems with interval uncertainties, reducing the need for extensive simulations.

Contribution

It develops two novel non-intrusive approaches employing Bayesian optimization for interval uncertainty propagation, providing probabilistic bounds and confidence intervals with minimal simulations.

Findings

Effective bounds estimation with fewer simulations

Probabilistic response description via Gaussian Processes

Confidence intervals improve decision-making

Abstract

Two non-intrusive uncertainty propagation approaches are proposed for the performance analysis of engineering systems described by expensive-to-evaluate deterministic computer models with parameters defined as interval variables. These approaches employ a machine learning based optimization strategy, the so-called Bayesian optimization, for evaluating the upper and lower bounds of a generic response variable over the set of possible responses obtained when each interval variable varies independently over its range. The lack of knowledge caused by not evaluating the response function for all the possible combinations of the interval variables is accounted for by developing a probabilistic description of the response variable itself by using a Gaussian Process regression model. An iterative procedure is developed for selecting a small number of simulations to be evaluated for updating this statistical model by using well-established acquisition functions and to assess the response bounds. In both approaches, an initial training dataset is defined. While one approach builds iteratively two distinct training datasets for evaluating separately the upper and lower bounds of the response variable, the other builds iteratively a single training dataset. Consequently, the two approaches will produce different bound estimates at each iteration. The upper and lower bound responses are expressed as point estimates obtained from the mean function of the posterior distribution. Moreover, a confidence interval on each estimate is provided for effectively communicating to engineers when these estimates are obtained for a combination of the interval variables for which no deterministic simulation has been run. Finally, two metrics are proposed to define conditions for assessing if the predicted bound estimates can be considered satisfactory.