Sparse Polynomial Chaos expansions using Variational Relevance Vector Machines

TL;DR

This paper introduces a novel sparse Bayesian learning method using Variational Relevance Vector Machines for Polynomial Chaos expansions, improving efficiency and accuracy in high-dimensional stochastic modeling with limited data.

Contribution

It presents a new variational Bayesian approach for sparse Polynomial Chaos expansions based on Relevance Vector Machines, suitable for high-dimensional data with few samples.

Findings

Effective in high-dimensional settings with limited data

Achieves sparsity levels comparable to compressive sensing

Validated on synthetic and real-world examples

Abstract

The challenges for non-intrusive methods for Polynomial Chaos modeling lie in the computational efficiency and accuracy under a limited number of model simulations. These challenges can be addressed by enforcing sparsity in the series representation through retaining only the most important basis terms. In this work, we present a novel sparse Bayesian learning technique for obtaining sparse Polynomial Chaos expansions which is based on a Relevance Vector Machine model and is trained using Variational Inference. The methodology shows great potential in high-dimensional data-driven settings using relatively few data points and achieves user-controlled sparse levels that are comparable to other methods such as compressive sensing. The proposed approach is illustrated on two numerical examples, a synthetic response function that is explored for validation purposes and a low-carbon steel plate with random Young's modulus and random loading, which is modeled by stochastic finite element with 38 input random variables.