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Uniqueness of MAP estimates for inverse problems under information field theory

Alex Alberts, Ilias Bilionis

TL;DR

This work addresses the issue of when MAP estimates are unique for inverse problems formulated in information field theory (IFT). By introducing a physics-informed prior with a model-trust parameter $\beta$, the authors study how data and physics interact and how model-form error can be detected through the learned $\beta$, including a Poisson-equation example. They derive gradient and Hessian expressions for the parameter posterior potential and establish conditions under which the inverse problem is weakly well-posed, especially in the free-theory (Gaussian) setting where the posterior is tractable. The key result is a theorem stating that, for field potentials linear in the parameters and under informative data, a zero-gradient implies a unique MAP, guaranteeing well-posedness; in the Poisson example, infinite $\beta$ corresponds to perfect physics, while finite $\beta$ signals model discrepancy. Overall, the paper provides a principled Bayesian framework for quantifying and detecting model-form error via the learned model trust, with implications for data-efficient physics-informed inference.

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.

Uniqueness of MAP estimates for inverse problems under information field theory

TL;DR

This work addresses the issue of when MAP estimates are unique for inverse problems formulated in information field theory (IFT). By introducing a physics-informed prior with a model-trust parameter , the authors study how data and physics interact and how model-form error can be detected through the learned , including a Poisson-equation example. They derive gradient and Hessian expressions for the parameter posterior potential and establish conditions under which the inverse problem is weakly well-posed, especially in the free-theory (Gaussian) setting where the posterior is tractable. The key result is a theorem stating that, for field potentials linear in the parameters and under informative data, a zero-gradient implies a unique MAP, guaranteeing well-posedness; in the Poisson example, infinite corresponds to perfect physics, while finite signals model discrepancy. Overall, the paper provides a principled Bayesian framework for quantifying and detecting model-form error via the learned model trust, with implications for data-efficient physics-informed inference.

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.
Paper Structure (14 sections, 10 theorems, 106 equations)

This paper contains 14 sections, 10 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Review of information field theory
  3. Setting and notation
  4. Construction of information field theory posteriors
  5. Discussion on model-form error
  6. Some properties of MAP estimates of IFT inverse problems
  7. Preliminaries
  8. Weakly well-posed inverse problems
  9. A free theory example
  10. Discussion and Conclusions
  11. Proof of lemma \ref{['thm:hessian']}
  12. Operator calculus and proof of Lemma \ref{['lem:covariance']}
  13. Primer on operator calculus for free theory
  14. Proof of Lemma \ref{['lem:covariance']}

Key Result

Proposition 3.1

Suppose that the Hessian of the parameter posterior potential, given by $\nabla_{\lambda}^2H(\lambda;d)$, is positive definite. If there exists $\lambda^*$ such that $\nabla_{\lambda}H(\lambda^*;d) = 0$, then the inverse problem is weakly well-posed and $\lambda^*$ is the unique MAP estimate.

Theorems & Definitions (29)

  • Definition 2.1: Free theory
  • Definition 3.1: Weakly well-posed inverse problem
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Definition 3.2: Conditional field expectations
  • Lemma 3.1
  • proof
  • Definition 3.3: Conditional field covariance
  • Lemma 3.2
  • ...and 19 more