A conditional normalizing flow for domain decomposed uncertainty quantification

TL;DR

This paper introduces a novel conditional normalizing flow model called cKRnet to improve uncertainty quantification across physical domains in PDE models, addressing density estimation bottlenecks in existing methods.

Contribution

The paper proposes a new conditional normalizing flow model, cKRnet, for efficient joint distribution estimation in domain decomposed uncertainty quantification, with theoretical analysis and numerical validation.

Findings

cKRnet effectively estimates conditional densities in uncertainty quantification.

CKR-DDUQ framework demonstrates improved accuracy and efficiency.

Numerical experiments validate the proposed method's convergence and performance.

Abstract

In this paper we present a conditional KRnet (cKRnet) based domain decomposed uncertainty quantification (CKR-DDUQ) approach to propagate uncertainties across different physical domains in models governed by partial differential equations (PDEs) with random inputs. This approach is based on the domain decomposed uncertainty quantification (DDUQ) method presented in [Q. Liao and K. Willcox, SIAM J. Sci. Comput., 37 (2015), pp. A103--A133], which suffers a bottleneck of density estimation for local joint input distributions in practice. In this work, we reformulate the required joint distributions as conditional distributions, and propose a new conditional normalizing flow model, called cKRnet, to efficiently estimate the conditional probability density functions. We present the general framework of CKR-DDUQ, conduct its convergence analysis, validate its accuracy and demonstrate its efficiency with numerical experiments.