Physics-informed information field theory for modeling physical systems with uncertainty quantification

Abstract

Data-driven approaches coupled with physical knowledge are powerful techniques to model engineering systems. The goal of such models is to efficiently solve for the underlying physical field through combining measurements with known physical laws. As many physical systems contain unknown elements, such as missing parameters, noisy measurements, or incomplete physical laws, this is widely approached as an uncertainty quantification problem. The common techniques to handle all of these variables typically depend on the specific numerical scheme used to approximate the posterior, and it is desirable to have a method which is independent of any such discretization. Information field theory (IFT) provides the tools necessary to perform statistics over fields that are not necessarily Gaussian. The objective of this paper is to extend IFT to physics-informed information field theory (PIFT) by encoding the functional priors with information about the physical laws which describe the field. The posteriors derived from this PIFT remain independent of any numerical scheme, and can capture multiple modes which allows for the solution of problems which are not well-posed. We demonstrate our approach through an analytical example involving the Klein-Gordon equation. We then develop a variant of stochastic gradient Langevin dynamics to draw samples from the field posterior and from the posterior of any model parameters. We apply our method to several numerical examples with various degrees of model-form error and to inverse problems involving non-linear differential equations. As an addendum, the method is equipped with a metric which allows the posterior to automatically quantify model-form uncertainty. Because of this, our numerical experiments show that the method remains robust to even an incorrect representation of the physics given sufficient data. We numerically demonstrate that the method correctly identifies when the physics cannot be trusted, in which case it automatically treats learning the field as a regression problem.