Can one hear the shape of a crack in a drum? Spectral fingerprints and a data-driven perspective

Abstract This paper investigates whether the shape of a crack in a membrane can be identified from the corresponding Laplace spectrum. To this end, analytical results are derived using the short-time asymptotic expansion of the heat trace, a fundamental spectral invariant. The analysis reveals that the existence, length and, under additional conditions, also the angle of a crack can be recovered, which allows for the reconstruction of certain crack shapes. Determining geometric parameters from the eigenfrequencies of a domain is a classical inverse problem, commonly referred to as the inverse spectral problem. In the second part of this work, a data-driven framework is presented, where a neural network is trained on simulated spectral data to approximate the inverse mapping. The simulations are carried out using isogeometric analysis, and the singularities at crack tips are resolved through graded mesh refinement. Error estimates are provided to ensure accurate simulation results. Numerical experiments confirm that the neural network reliably reconstructs crack parameters from spectral input.