Causality and Bayesian Network PDEs for multiscale representations of porous media

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 momentindependent 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.

Explainable and trustworthy artificial intelligence for correctable modeling in chemical sciences

“…Such predictions are nonintuitive. While previous work found that parametric-based microscale uncertainties can be dampened in multiscale models (43), the results of this work will generalize to any models where finescale simulations (such as DFT) are sparse or the macroscale QoIs can be made proportional to the microscale properties. In the next section, we show that Eq.…”

Distribution‐Based Global Sensitivity Analysis in Hydrology

“…In that case, singular‐value decomposition techniques, such as a truncated Karhunen‐Loève transformation, are used to approximate p with a set of M ( M ≪ M par ) mutually uncorrelated identically distributed random variables ξ ={ ξ 1 ,…, ξ M }. Alternatively, if p ={ p 1 ,…, p M } represents M par = M correlated random variables (rather than random fields), then the Rosenblatt transform (e.g., section 4.1 in Um et al., ) maps p onto a set of random variables ξ ={ ξ 1 ,…, ξ M } that are independent and identically distributed on the interval (0,1).…”

“…This renders it inapplicable, except as an uncontrollable approximation, to hydrologic systems, whose parameters are often correlated random fields and are cross-correlated with each other. Such problems call for the use of moment-independent GSA techniques, including distribution-based GSA (Borgonovo, 2007), a PCE-based method of Caniou and Sudret (2011), and Bayesian nets (Um et al, 2019). Yet hydrologic applications of moment-independent GSA accelerated by PCE-based surrogates remain scarce (e.g., Rajabi et al, 2015).…”

Characterizing the Influence of Multiple Uncertainties on Predictions of Contaminant Discharge in Groundwater Within a Lagrangian Stochastic Formulation

“…Note that in case of correlated input parameters one has to preliminary transform p onto a set of independent random variables to derive the PCE approximation (e.g., Section 4.1 in Um et al, 2019).…”