  In this paper, we present a novel approach for detecting the discontinuity
interfaces of a discontinuous function. This approach leverages Graph-Informed
Neural Networks (GINNs) and sparse grids to address discontinuity detection
also in domains of dimension larger than 3. GINNs, trained to identify troubled
points on sparse grids, exploit graph structures built on the grids to achieve
efficient and accurate discontinuity detection performances. We also introduce
a recursive algorithm for general sparse grid-based detectors, characterized by
convergence properties and easy applicability. Numerical experiments on
functions with dimensions n = 2 and n = 4 demonstrate the efficiency and robust
generalization of GINNs in detecting discontinuity interfaces. Notably, the
trained GINNs offer portability and versatility, allowing integration into
various algorithms and sharing among users.
