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

Journal Article · Wed Oct 19 00:00:00 EDT 2022 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2022.111691· OSTI ID:2417951

Wentz, Jacqueline ^[1]; Doostan, Alireza ^[1]

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1. Univ. of Colorado, Boulder, CO (United States)

Here, 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.

• Research Organization: Univ. of Colorado, Boulder, CO (United States)
• Sponsoring Organization: US Air Force Office of Scientific Research (AFOSR); USDOE; USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: NA0003962
• OSTI ID: 2417951
• Alternate ID(s): OSTI ID: 1896327; OSTI ID: 2997176
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 472; ISSN 0021-9991
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
Compressed sensing
Computer Science
Nonlinear approximation
Physics
Polynomial chaos expansion
Sparse approximation
Uncertainty quantification