Enforcing Boundary Conditions on Physical Fields in Bayesian Inversion

TL;DR

This paper introduces a method to enforce boundary conditions on physical fields in Bayesian inverse problems by modifying the covariance kernel, improving the physical plausibility of inferred fields in computational mechanics.

Contribution

It presents a novel approach to incorporate boundary conditions directly into the statistical model for latent fields in Bayesian inversion.

Findings

Enforcing boundary conditions improves boundary behavior of inferred fields.

Modified covariance kernels ensure all realizations satisfy boundary constraints.

Method applied successfully to infer eddy viscosity in Navier-Stokes equations.

Abstract

Inverse problems in computational mechanics consist of inferring physical fields that are latent in the model describing some observable fields. For instance, an inverse problem of interest is inferring the Reynolds stress field in the Navier--Stokes equations describing mean fluid velocity and pressure. The physical nature of the latent fields means they have their own set of physical constraints, including boundary conditions. The inherent ill-posedness of inverse problems, however, means that there exist many possible latent fields that do not satisfy their physical constraints while still resulting in a satisfactory agreement in the observation space. These physical constraints must therefore be enforced through the problem formulation. So far there has been no general approach to enforce boundary conditions on latent fields in inverse problems in computational mechanics, with these constraints often simply ignored. In this work we demonstrate how to enforce boundary conditions in Bayesian inversion problems by choice of the statistical model for the latent fields. Specifically, this is done by modifying the covariance kernel to guarantee that all realizations satisfy known values or derivatives at the boundary. As a test case the problem of inferring the eddy viscosity in the Reynolds-averaged Navier--Stokes equations is considered. The results show that enforcing these constraints results in similar improvements in the output fields but with latent fields that behave as expected at the boundaries.