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. The applicability of PCE to high-dimensional problems is limited by poor scalability, as the number of basis functions grows exponentially with the number of parameters. In this paper, we address this challenge by combining PCE with ideas from tractable probabilistic circuits, resulting in 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 multilayer perceptrons (MLPs), while retaining PCE's ability to compute exact statistical inferences via simple forward passes. In contrast, such computations in MLPs require costly and often inaccurate approximations, such as Monte Carlo integration.