Causality and Bayesian Network PDEs for multiscale representations of porous media

Kimoon Um, Eric J. Hall, Markos A. Katsoulakis, Daniel M. Tartakovsky

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Microscopic (pore-scale) properties of porous media affect and often determine their macroscopic (continuum- or Darcy-scale) counterparts. Understanding the relationship between processes on these two scales is essential to both the derivation of macroscopic models of, e.g., transport phenomena in natural porous media, and the design of novel materials, e.g., for energy storage. Microscopic properties exhibit complex statistical correlations and geometric constraints that present challenges for the estimation of macroscopic quantities of interest (QoIs), e.g., in the context of global sensitivity analysis (GSA) of macroscopic QoIs with respect to microscopic material properties. We present a systematic way of building correlations into stochastic multiscale models through Bayesian Networks. The proposed framework allows us to construct the joint probability density function (PDF) of model parameters through causal relationships that are informed by domain knowledge and emulate engineering processes, e.g., the design of hierarchical nanoporous materials. These PDFs also serve as input for the forward propagation of parametric uncertainty thereby yielding a Bayesian Network PDE. To assess the impact of causal relationships and microscale correlations on macroscopic material properties, we propose a moment-independent GSA and corresponding effect rankings for Bayesian Network PDEs, based on the differential Mutual Information, that leverage the structure of Bayesian Networks and account for both correlated inputs and complex non-Gaussian (skewed, multimodal) QoIs. Our findings from numerical experiments, which feature a non-intrusive uncertainty quantification workflow, indicate two practical outcomes. First, the inclusion of correlations through structured priors based on causal relationships informed by domain knowledge impacts predictions of QoIs and has important implications for engineering design. Second, structured priors with non-trivial correlations yield different effect rankings than independent priors; these rankings are more consistent with the anticipated physics.

Original languageEnglish
Pages (from-to)658-678
Number of pages21
JournalJournal of Computational Physics
Early online date7 Jun 2019
DOIs
Publication statusPublished - Oct 2019

Keywords

  • Bayesian networks
  • Causality
  • Porous media
  • Energy storage
  • Global Sensitivity Analysis
  • Mutual information

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