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.

翻译：本文提出了一种新颖的方法，用于检测不连续函数的不连续界面。该方法利用图信息神经网络（GINNs）和稀疏网格，可在维度大于3的区域中解决不连续检测问题。通过训练GINNs识别稀疏网格上的问题点，其利用基于网格构建的图结构，实现了高效准确的不连续检测性能。我们还提出了一种通用的稀疏网格检测器递归算法，该算法具有收敛性且易于应用。在维度n=2和n=4的函数上进行的数值实验证明，GINNs在不连续界面检测方面具有高效性和鲁棒泛化能力。值得注意的是，训练后的GINNs具备可移植性和多功能性，可集成至多种算法中并在用户间共享。