Mathematics Meets AI for Inverse Problems: Tracking Moving Scatterers and Beyond
ABSTRACT
Inverse problems aim to identify the geometric and material properties of an object from measurement data. The complexity of realistic configurations and measurement settings can make such problems challenging, motivating the integration of mathematical analysis with AI-based methods. In this talk, we apply this approach to the inverse scattering problem of tracking the location and orientation of a moving target, using far-field measurements from a single incident field. We leverage analytical properties of the far-field data together with Bayesian optimization to efficiently track two- and three-dimensional sound-soft scatterers while reducing the number of objective function evaluations. When the target shape is unknown, machine learning is used to infer the shape from measurement data. Once the shape is identified, the target configuration can be recovered from subsequent far-field measurements without repeatedly measuring the target shape or retraining the model, by exploiting its analytical properties under translations and rotations. Numerical experiments with randomly generated shapes and trajectories demonstrate the proposed approach. This work highlights the potential of integrating mathematical and AI-based approaches into broader classes of inverse problems.
