Phi-ML meets Engineering seminar series: Feynman path integral formulation of inverse problems: Lifting the assumption that the model is correct

Free Event series

Phi-ML meets Engineering seminar series

Introduction

This bi-monthly seminar series explores real-world applications of physics-informed machine learning (Φ-ML) methods to the engineering practice. They cover a wide range of topics, offering a cross-sectional view of the state of the art on Φ-ML research, worldwide. Participants have the opportunity to hear from leading researchers and learn about the latest developments in this emerging field. These seminars also offer the chance to identify and spark collaboration opportunities.

About the event

In inverse problems, we want to reconstruct physical fields (e.g., velocity, and pressure) and identify unknown model parameters from incomplete noisy observations. One formulates inverse problems following a Bayesian paradigm, assuming that the underlying forward model (usually an ordinary or partial differential equation) is correct. Using Bayesian asymptotic theory, one can show that the posterior over the model parameters collapses to a Dirac delta centered about the maximum likelihood estimate. When the model is correct, this collapse is a desired feature, i.e., in the infinite data limit, one recovers the parameters' actual value. However, the collapse occurs even when the model is wrong. In other words, the typical inverse problem formulation is not robust to model misspecification. In this talk, I will use the formalism of Feynman path integrals to relax the assumption that the forward model is correct. Specifically, I will construct a prior probability measure over the function space of physical fields such that fields that satisfy the assumed differential equations become more likely. The prior includes a learnable parameter corresponding to the trust one puts in the model. I will then condition this prior probability measure on the available data. The resulting posterior includes an intractable normalization constant that needs to be treated with care. I will discuss two approaches for characterizing the posterior: nested variational inference and preconditioned stochastic gradient Langevin dynamics. I will provide numerical and theoretical evidence demonstrating the use of the trust parameter. I will demonstrate the use of the methodology for the reconstruction of hemodynamic flows.

Watch on demand: Recording will be uploaded after the seminar.