A hierarchical neural hybrid method for failure probability estimation

TL;DR

This paper introduces a hierarchical neural hybrid method that significantly reduces the computational cost of estimating failure probabilities in high-dimensional PDE-governed systems by adaptively using multifidelity surrogates.

Contribution

The paper presents a novel hierarchical neural hybrid approach that efficiently combines multifidelity surrogates to overcome the curse of dimensionality in failure probability estimation.

Findings

HNH method reduces PDE solves from over a million to a few thousand for high accuracy.

Theoretical analysis confirms efficiency gains of the HNH approach.

Numerical experiments demonstrate the method's effectiveness in complex high-dimensional problems.

Abstract

Failure probability evaluation for complex physical and engineering systems governed by partial differential equations (PDEs) are computationally intensive, especially when high-dimensional random parameters are involved. Since standard numerical schemes for solving these complex PDEs are expensive, traditional Monte Carlo methods which require repeatedly solving PDEs are infeasible. Alternative approaches which are typically the surrogate based methods suffer from the so-called ``curse of dimensionality'', which limits their application to problems with high-dimensional parameters. For this purpose, we develop a novel hierarchical neural hybrid (HNH) method to efficiently compute failure probabilities of these challenging high-dimensional problems. Especially, multifidelity surrogates are constructed based on neural networks with different levels of layers, such that expensive highfidelity surrogates are adapted only when the parameters are in the suspicious domain. The efficiency of our new HNH method is theoretically analyzed and is demonstrated with numerical experiments. From numerical results, we show that to achieve an accuracy in estimating the rare failure probability (e.g., $10^{-5}$), the traditional Monte Carlo method needs to solve PDEs more than a million times, while our HNH only requires solving them a few thousand times.