Sparse Polynomial Chaos expansions using Variational Relevance Vector Machines
This work addresses computational bottlenecks in uncertainty quantification for engineering applications, offering an incremental improvement in sparse modeling methods.
The authors tackled the challenge of computational efficiency and accuracy in non-intrusive Polynomial Chaos modeling by developing a sparse Bayesian learning technique based on Relevance Vector Machines and Variational Inference, achieving user-controlled sparsity comparable to compressive sensing with few data points in high-dimensional settings, as demonstrated on a synthetic function and a steel plate model with 38 random variables.
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.