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.

本文介绍了一种新颖的方法来检测不连续函数的不连续界面，该方法利用了基于图的神经网络 (GINNs) 和稀疏网格来解决大于 3 维的领域中的不连续检测问题。我们还引入了一种递归算法，用于基于稀疏网格的通用检测器，具有收敛性和易用性。在维数为 2 和 4 的函数上进行的数值实验证明了 GINNs 在检测不连续界面方面的效率和鲁棒泛化性。值得注意的是，训练的 GINNs 具有可移植性和通用性，可集成到各种算法中并在用户之间共享。

基于图信息的稀疏网格间断检测神经网络

Graph-Informed Neural Networks for Sparse Grid-Based Discontinuity  Detectors

Structured kernel interpolation (SKI) accelerates Gaussian process (GP)
inference by interpolating the kernel covariance function using a dense grid of
inducing points, whose corresponding kernel matrix is highly structured and
thus amenable to fast linear algebra. Unfortunately, SKI scales poorly in the
dimension of the input points, since the dense grid size grows exponentially
with the dimension. To mitigate this issue, we propose the use of sparse grids
within the SKI framework. These grids enable accurate interpolation, but with a
number of points growing more slowly with dimension. We contribute a novel
nearly linear time matrix-vector multiplication algorithm for the sparse grid
kernel matrix. Next, we describe how sparse grids can be combined with an
efficient interpolation scheme based on simplices. With these changes, we
demonstrate that SKI can be scaled to higher dimensions while maintaining
accuracy.

本文提出了在结构核插值框架中使用稀疏网格的方法，改进了高斯过程推理的可扩展性。这种方法结合了线性代数和基于单纯形的高维插值方案，使得 SKI 能够在更高维度下可靠地实现精确的推理。