Abstract

Information field theory (IFT) is Bayesian statistics for fields (a.k.a. functions of space and time). It uses the same mathematics found in statistical field theory and quantum field theory, namely functional integration or Feynman path integrals. IFT starts by imposing a prior probability measure over the space of physical fields (e.g., temperature, pressure, strain, stress). Then, one constructs a likelihood function that models the measurement process to connect the fields to the available data. Finally, one uses Bayes’ rule to build a posterior over the space of physical fields, which they proceed to characterize either analytically (see Feynman diagrams) or numerically. IFT has been used successfully in various field reconstruction problems, primarily astrophysical applications. In this talk, we will discuss how IFT can be used to perform uncertainty quantification tasks in physical problems governed by ordinary and partial differential equations. We will show how one can 1) use knowledge of the governing equations to construct suitable prior measures over the space of fields; 2) sample from the fields’ posterior numerically via advanced Markov chain Monte Carlo and variational inference without the need to call a numerical solver; and 3) sample from the posterior of any physical parameters, initial conditions, boundary conditions, and source terms without the need to evaluate a normalization constant. The method offers several potential advantages compared to traditional uncertainty quantification techniques. The approach has a mechanism for quantifying the model-form uncertainty. It naturally fuses data from multiple modalities and elegantly deals with ill-posed problems (e.g., missing boundary conditions).