Uniqueness of MAP estimates for inverse problems under information field theory

TL;DR

This paper investigates the conditions for the uniqueness of MAP estimates in inverse problems formulated with information field theory, emphasizing model-form error detection and the role of physics-informed priors.

Contribution

It introduces conditions for MAP estimate uniqueness in IFT inverse problems and proposes a method to detect model-form error by learning the model trust parameter.

Findings

Correct models yield infinite trust values.

Physics closer to ground truth increase model trust.

Model-form error reduces the trust parameter.

Abstract

Information field theory (IFT) is an emerging technique for posing infinite-dimensional inverse problems using the mathematics found in quantum field theory. Under IFT, the field inference task is formulated in a Bayesian setting where the probability measures are defined by path integrals. We derive conditions under which IFT inverse problems have unique maximum a posterioi estimates, placing a special focus on the problem of identifying model-form error. We define physics-informed priors over fields, where a parameter, called the model trust, measures our belief in the physical model. Smaller values of trust cause the prior to diffuse, representing a larger degree of uncertainty about the physics. To detect model-form error, we learn the trust as part of the inverse problem and study the limiting behavior. We provide an example where the physics are assumed to be the Poisson equation and study the effect of model-form error on the model trust. We find that a correct model leads to infinite trust, and under model-form error, physics that are closer to the ground truth lead to larger values of the trust.