Bayesian inference of an uncertain generalized diffusion operator

TL;DR

This paper introduces a Bayesian framework for inferring an uncertain generalized diffusion operator in a differential equation, using eigendecomposition parametrization to handle sparse data in a contaminant transport model.

Contribution

It proposes a novel Bayesian inverse problem formulation for infinite-dimensional operators, incorporating prior constraints and qualitative system information to improve inference with limited data.

Findings

Effective operator inference with sparse data demonstrated

Prior constraints improve inference accuracy

Dependence on observation frequency and data type analyzed

Abstract

This paper defines a novel Bayesian inverse problem to infer an infinite-dimensional uncertain operator appearing in a differential equation, whose action on an observable state variable affects its dynamics. Inference is made tractable by parametrizing the operator using its eigendecomposition. The plausibility of operator inference in the sparse data regime is explored in terms of an uncertain, generalized diffusion operator appearing in an evolution equation for a contaminant's transport through a heterogeneous porous medium. Sparse data are augmented with prior information through the imposition of deterministic constraints on the eigendecomposition and the use of qualitative information about the system in the definition of the prior distribution. Limited observations of the state variable's evolution are used as data for inference, and the dependence on the solution of the inverse problem is studied as a function of the frequency of observations, as well as on whether or not the data is collected as a spatial or time series.