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：本文提出了一种用于检测不连续函数间断界面的新方法。该方法利用图信息神经网络（GINNs）与稀疏网格，可在维度大于3的域中实现不连续检测。通过在稀疏网格上训练GINNs识别问题点，并利用网格构建的图结构，实现了高效且准确的不连续检测性能。我们同时提出了一种适用于通用稀疏网格检测器的递归算法，该算法具有收敛特性且易于实现。针对n=2和n=4维函数的数值实验表明，GINNs在检测不连续界面方面具有高效性与鲁棒泛化能力。值得注意的是，训练后的GINNs具有良好的可移植性与多用途性，可集成至不同算法中并在用户间共享。