New preprint on crack identification in membranes using spectral data

20 Januar 2026

Our new preprint “Can one hear the shape of a crack in a drum? Spectral fingerprints and a data-driven perspective” by Thomas Apel, Serge Nicaise, and Philipp Zilk is now available on the French open archive HAL (Hyper Articles en Ligne).

In this work, we address an inverse spectral problem and investigate whether geometric properties of a crack in a membrane can be recovered from the associated Laplace spectrum. By means of short-time asymptotic expansions of the heat trace, we derive analytical results showing that the existence and length of a crack can be identified and that, under additional assumptions, further geometric parameters can be recovered from the corresponding eigenfrequencies.

Complementing the analytical study, the second part of the paper introduces a data-driven framework. A neural network is trained on accurately simulated spectral data to approximate the inverse mapping from eigenfrequencies to crack parameters. The simulations are performed using isogeometric analysis, with graded mesh refinement to properly resolve the singularities at crack tips. Error estimates ensure the reliability of the numerical data, and numerical experiments demonstrate that the trained network can robustly reconstruct crack parameters from spectral input.

Thomas Apel, Serge Nicaise, Philipp Zilk. Can one hear the shape of a crack in a drum? Spectral fingerprints and a data-driven perspective. 2025. ⟨hal-05415870⟩