Deep Polynomial Chaos Expansion

Abstract

Polynomial chaos expansion (PCE) is a classical and widely used surrogate modeling technique in physical simulation and uncertainty quantification. By taking a linear combination of a set of basis polynomials - orthonormal with respect to the distribution of uncertain input parameters - PCE enables tractable inference of key statistical quantities, such as (conditional) means, variances, covariances, and Sobol sensitivity indices, which are essential for understanding the modeled system and identifying influential parameters and their interactions. As the number of basis functions grows exponentially with the number of parameters, PCE does not scale well to high-dimensional problems. We address this challenge by combining PCE with ideas from probabilistic circuits, resulting in the deep polynomial chaos expansion (DeepPCE) - a deep generalization of PCE that scales effectively to high-dimensional input spaces. DeepPCE achieves predictive performance comparable to that of multi-layer perceptrons (MLPs), while retaining PCE's ability to compute exact statistical inferences via simple forward passes.