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

Detecting discontinuity interfaces of discontinuous functions is a challenging task with significant implications across various scientific and engineering applications. Identifying these interfaces is particularly critical for functions with a high-dimensional domain, as their discontinuities can significantly influence the behavior of numerical methods and simulations; for example, within the realm of uncertainty quantification, where the smoothness of the target function plays a fundamental role in the use of stochastic collocation methods. Specifically, the knowledge of discontinuity interfaces enables the partitioning of the function domain into regions of smoothness, a crucial factor in improving the performance of numerical methods (e.g., see [17]). Other examples of discontinuity detection applications include signal processing, nonlinear partial differential equation (PDE) simulations, investigations of phase transitions in physical systems [14], and change-point analyses in geology or biology, to name a few [30]. The central objective of most discontinuity detection methods is to identify the position of discontinuities in the function domain using function evaluations on sets of points. Over the last few decades, progresses has been made in discontinuity detection, leading to the development of various algorithms. Notable works, such as [3, 2, 16, 35], have introduced significant contributions in this field. In particular, [3] introduced a polynomial annihilation edge detection method designed for piece-wise smooth functions with low-dimensional domains (n 2). This method identifies discontinuous interfaces by reconstructing jump functions based on a set of function evaluations.