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

TL;DR

This paper introduces Graph-Instructed Neural Networks (GINNs) combined with sparse grids to efficiently detect discontinuity interfaces in high-dimensional functions, demonstrating robustness and portability in numerical experiments.

Contribution

The paper proposes a novel GINN-based method for discontinuity detection in high-dimensional sparse grid domains, including a recursive algorithm with proven convergence.

Findings

GINNs effectively detect discontinuities in functions up to 4 dimensions.

The method shows robust generalization and portability across different algorithms.

Numerical experiments confirm efficiency and accuracy of the proposed approach.

Abstract

In this paper, we present a novel approach for detecting the discontinuity interfaces of a discontinuous function. This approach leverages Graph-Instructed 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 properties of GINNs in detecting discontinuity interfaces. Notably, the trained GINNs offer portability and versatility, allowing integration into various algorithms and sharing among users.