Sequential Bayesian experimental design for estimation of extreme-event probability in stochastic dynamical systems

TL;DR

This paper introduces a sequential Bayesian experimental design method using variational heteroscedastic Gaussian process regression to efficiently estimate extreme-event probabilities in complex stochastic systems with expensive response functions.

Contribution

The paper develops a novel sequential Bayesian design approach that accounts for heteroscedastic noise, improving efficiency in estimating rare event probabilities in high-dimensional stochastic systems.

Findings

Validated on synthetic problems with artificial stochastic responses.

Successfully applied to estimate extreme ship motion probability in irregular waves.

Reduces sampling requirements compared to traditional methods.

Abstract

We consider an input-to-response (ItR) system characterized by (1) parameterized input with a known probability distribution and (2) stochastic ItR function with heteroscedastic randomness. Our purpose is to efficiently quantify the extreme response probability when the ItR function is expensive to evaluate. The problem setup arises often in physics and engineering problems, with randomness in ItR coming from either intrinsic uncertainties (say, as a solution to a stochastic equation) or additional (critical) uncertainties that are not incorporated in a low-dimensional input parameter space (as a result of dimension reduction applied to the original high-dimensional input space). To reduce the required sampling numbers, we develop a sequential Bayesian experimental design method leveraging the variational heteroscedastic Gaussian process regression (VHGPR) to account for the stochastic ItR, along with a new criterion to select the next-best samples sequentially. The validity of our new method is first tested in two synthetic problems with the stochastic ItR functions defined artificially. Finally, we demonstrate the application of our method to an engineering problem of estimating the extreme ship motion probability in irregular waves, where the uncertainty in ItR naturally originates from standard wave group parameterization, which reduces the original high-dimensional wave field into a two-dimensional parameter space.