Graph-instructed neural networks for sparse grid-based discontinuity detectors

In this paper, we present a novel approach for detecting the discontinuity interfaces of a discontinuous function. This approach leverages Graph-Instructed Neural Networks (GINNs) and sparse grids to address discontinuity detection even 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 performance. We also introduce a recursive algorithm for general sparse grid-based detectors, characterized by convergence properties and ease of applicability. Numerical experiments on functions with dimensions n=2 and n=4 demonstrate the efficiency and robust generalization properties of GINNs in detecting discontinuity interfaces; test cases with n=6 and n=8 show the applicability of the method when the discontinuity interface presents specific structures. Notably, the trained GINNs offer portability and versatility, allowing integration into various algorithms and sharing among users.