Causality and Bayesian network PDEs for multiscale representations of porous media

TL;DR

This work develops a Bayesian-network PDE framework to couple pore-scale geometry with Darcy-scale transport in hierarchical nanoporous media, enabling causal, physics-informed uncertainty propagation through a two-scale homogenization. It combines Rosenblatt decorrelation, truncated gPCE surrogates, and KDE-based density estimation (facilitated by DAKOTA) to perform moment-independent global sensitivity analysis via differential mutual information on non-Gaussian QoIs. The study demonstrates that incorporating pore-scale causal constraints yields nontrivial correlations and materially impacts predicted macroscopic quantities, with rankings that align with physical intuition in some regimes and reveal new insights in others. The framework supports flexible alternative designs and model-form uncertainty, highlighting the value of learning Bayesian-network structures from data to robustly inform material design and reliability assessments in energy storage and related applications.

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. Most microscopic properties exhibit complex statistical correlations and geometric constraints, which presents 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. This allows us to construct the joint probability density function (PDF) of model parameters through causal relationships that emulate engineering processes, e.g., the design of hierarchical nanoporous materials. Such PDFs also serve as input for the forward propagation of parametric uncertainty; our findings indicate that the inclusion of causal relationships impacts predictions of macroscopic QoIs. To assess the impact of correlations and causal relationships between microscopic parameters on macroscopic material properties, we use a moment-independent GSA based on the differential mutual information. Our GSA accounts for the correlated inputs and complex non-Gaussian QoIs. The global sensitivity indices are used to rank the effect of uncertainty in microscopic parameters on macroscopic QoIs, to quantify the impact of causality on the multiscale model's predictions, and to provide physical interpretations of these results for hierarchical nanoporous materials.

Figures (11)

• Figure 1: A hierarchical nanoporous material UmZhangKatsoulakisEtAl:2017aa exhibiting horizontally oriented nanotunnels through mesopores connected by a series of vertically oriented nanotubes. The porous media volume $\mathcal{V}$ (left) consists of a periodic arrangement of unit cells ${\breve{\mathcal{V}}}$ (right) with pore space ${\breve{\mathcal{P}}}$ and fluid-solid interface ${\breve{\Gamma}}$. The parameters $\{R, \theta, d, l\}$ describing the nanopore features are constrained by the geometry of ${\breve{\mathcal{V}}}$.
• Figure 2: A Bayesian network describing the components of the full statistical model $P$, in \ref{['eq:statistical-model']}, for the multiscale porous media system takes into account the joint PDF $P(\boldsymbol{\Theta})$ of the input parameters, the PDF $P(\boldsymbol{X} \mid \boldsymbol{\Theta})$ of the upscaling variable that maps pore-scale properties to Darcy-scale variables, and the PDF $P(U \mid \boldsymbol{X})$ related to Darcy-scale QoIs. This figure and other Bayesian networks are produced using scipy:daft.
• Figure 3: A Bayesian network describing the components of the full statistical model under the assumption of independent priors on pore-scale features $\boldsymbol{\Theta} = (\Theta_1, \dots, \Theta_n)$. The flat structure of the $\boldsymbol{\Theta}$ component in the model above contrasts with the rich structure of the Bayesian network in \ref{['fig:conditional-d-and-l']} that captures causal relationships among the pore-scale features in order to ensure sampling geometries consistent with the hierarchical nanoporous material in \ref{['fig:pore-structure']} over the physically relevant hyperparameter ranges in \ref{['tab:extended-range']}.
• Figure 4: The rich structures of the Bayesian network above, representing the probabilistic model $P_1$ in \ref{['eq:P_1']}, encodes causal relationships arising from structural constraints (cf. \ref{['fig:upperbound_d']}) that are absent in the model $P_0$ in \ref{['eq:P_0']} with independent priors in \ref{['fig:general-model-indpriors']}. In this Bayesian network, conditional dependencies among the variables $\boldsymbol{\Theta}$ induce various correlation structures that depend on the selected hyperparameters (cf. the correlation structure for model $P_1$ over the narrow range of hyperparameters in \ref{['fig:corr-exB-narrow']} vs. the physical range in \ref{['fig:corr-exB-extended']}).
• Figure 5: The conditional distribution of $\Theta_d$ and $\Theta_l$ in \ref{['eq:Theta_d-given-R-theta', 'eq:Theta_l-given-R-theta-d']} arises from geometric constraints that arise naturally when considering the hierarchical nanopore structure in \ref{['fig:pore-structure']}. Above, we illustrate that the nanotube radius $d/2$ cannot exceed $R\cos\theta$, half the width of the unit cell ${\breve{\mathcal{V}}}$ resulting in \ref{['eq:Theta_d-given-R-theta']}. Further, to exclude gaps between vertical mesopores that are less than zero, $l > 2R - \sqrt{4R^2-d^2}$ from the right triangle in the diagram above resulting in \ref{['eq:Theta_l-given-R-theta-d']}.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4