  Detecting hidden geometrical structures from surface measurements under
electromagnetic, acoustic, or mechanical loading is the goal of noninvasive
imaging techniques in medical and industrial applications. Solving the inverse
problem can be challenging due to the unknown topology and geometry, the
sparsity of the data, and the complexity of the physical laws. Physics-informed
neural networks (PINNs) have shown promise as a simple-yet-powerful tool for
problem inversion, but they have yet to be applied to general problems with a
priori unknown topology. Here, we introduce a topology optimization framework
based on PINNs that solves geometry detection problems without prior knowledge
of the number or types of shapes. We allow for arbitrary solution topology by
representing the geometry using a material density field that approaches binary
values thanks to a novel eikonal regularization. We validate our framework by
detecting the number, locations, and shapes of hidden voids and inclusions in
linear and nonlinear elastic bodies using measurements of outer surface
displacement from a single mechanical loading experiment. Our methodology opens
a pathway for PINNs to solve various engineering problems targeting geometry
optimization.
