GenMod: A generative modeling approach for spectral representation of PDEs with random inputs

TL;DR

This paper introduces GenMod, a novel generative modeling approach for efficiently quantifying uncertainty in high-dimensional PDE systems with limited solution evaluations, leveraging spectral representations and compressed sensing techniques.

Contribution

The paper presents a new method combining generative models with spectral PDE solutions to improve uncertainty quantification with fewer samples, including theoretical recovery guarantees.

Findings

GenMod outperforms sparsity-promoting methods at small sample sizes.

The approach effectively predicts polynomial chaos coefficients in high-dimensional PDEs.

Theoretical analysis supports the recovery guarantees for the proposed method.

Abstract

We propose a method for quantifying uncertainty in high-dimensional PDE systems with random parameters, where the number of solution evaluations is small. Parametric PDE solutions are often approximated using a spectral decomposition based on polynomial chaos expansions. For the class of systems we consider (i.e., high dimensional with limited solution evaluations) the coefficients are given by an underdetermined linear system in a regression formulation. This implies additional assumptions, such as sparsity of the coefficient vector, are needed to approximate the solution. Here, we present an approach where we assume the coefficients are close to the range of a generative model that maps from a low to a high dimensional space of coefficients. Our approach is inspired be recent work examining how generative models can be used for compressed sensing in systems with random Gaussian measurement matrices. Using results from PDE theory on coefficient decay rates, we construct an explicit generative model that predicts the polynomial chaos coefficient magnitudes. The algorithm we developed to find the coefficients, which we call GenMod, is composed of two main steps. First, we predict the coefficient signs using Orthogonal Matching Pursuit. Then, we assume the coefficients are within a sparse deviation from the range of a sign-adjusted generative model. This allows us to find the coefficients by solving a nonconvex optimization problem, over the input space of the generative model and the space of sparse vectors. We obtain theoretical recovery results for a Lipschitz continuous generative model and for a more specific generative model, based on coefficient decay rate bounds. We examine three high-dimensional problems and show that, for all three examples, the generative model approach outperforms sparsity promoting methods at small sample sizes.