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A conditional normalizing flow for domain decomposed uncertainty quantification

Sen Li, Ke Li, Yu Liu, Qifeng Liao

TL;DR

This work reformulates the required joint distributions as conditional distributions, and proposes a new conditional normalizing flow model, called cKRnet, to efficiently estimate the conditional probability density functions.

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.

A conditional normalizing flow for domain decomposed uncertainty quantification

TL;DR

This work reformulates the required joint distributions as conditional distributions, and proposes a new conditional normalizing flow model, called cKRnet, to efficiently estimate the conditional probability density functions.

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.

Paper Structure

This paper contains 18 sections, 4 theorems, 70 equations, 18 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setup and domain decomposed uncertainty quantification
  3. Settings for non-overlapping domain decomposition
  4. The domain-decomposed uncertainty quantification (DDUQ) approach
  5. Conditional KRnet and conditional density estimation
  6. Conditional affine coupling layers
  7. Conditional KRnet (cKRnet) structure
  8. cKRnet for conditional density estimation
  9. Performance of cKRnet
  10. Conditional KRnet for DDUQ (CKR-DDUQ)
  11. CKR-DDUQ Algorithm
  12. Convergence analysis
  13. Numerical study
  14. Two-component diffusion problem with 28 parameters
  15. Three-component diffusion problem with 42 parameters
  16. ...and 3 more sections

Key Result

Lemma 1

If the DD-convergence condition is satisfied for all $\xi\in \Gamma$, then for any given number $a$, $\tilde{P}(y_i \leq a)$=$P(y_i \leq a)$.

Figures (18)

  • Figure 1: Relative errors of cKRnet (CKR) and KRnet (KR) for the Gaussian mixture distribution test problem.
  • Figure 2: Illustration of the spatial domain with two components.
  • Figure 3: Maximum of the error indicator on subdomain $D_1$ ($\max_{s=1:{N_{\rm {off}}}}\|\tau^{k+1}_1(\xi^{(s)})-\tau^k_1(\xi^{(s)})\|_{\infty}$) and that on subdomain $D_2$ ($\max_{s=1:{N_{\rm {off}}}}\|\tau^{k+1}_2(\xi^{(s)})-\tau^k_2(\xi^{(s)})\|_{\infty}$) for the coupling surrogates, two-component diffusion test problem.
  • Figure 4: Average CKR-DDUQ and KR-DDUQ errors in mean and variance estimates for each output $y_i$ ($\mathbf{E}(\epsilon_i)$ and $\mathbf{E}(\eta_i))$, $i=1,2$, two-component diffusion test problem.
  • Figure 5: PDFs of the outputs of interest, two-component diffusion test problem.
  • ...and 13 more figures

Theorems & Definitions (8)

  • definition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof