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，用于检测间断函数的间断界面。该方法利用图信息神经网络（GINNs）和稀疏网格，实现维度大于3的区域中的间断检测。通过训练GINNs识别稀疏网格上的困难点，并利用网格上构建的图结构，实现了高效且准确的间断检测性能。我们还引入了一种适用于一般稀疏网格检测器的递归算法，该算法具有收敛性且易于应用。针对维度n=2和n=4的函数进行的数值实验表明，GINNs在检测间断界面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具有可移植性和通用性，可集成到各种算法中并在用户间共享。