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

TL;DR

The paper tackles high-dimensional discontinuity detection by learning discontinuity detectors on sparse-grid graphs. It introduces Graph-Instructed Neural Networks (GINNs) trained on synthetic discontinuity datasets and embeds them in a recursive sparse-grid detector framework that converges under exact detectors and remains efficient in dimensions $n>3$. Empirical results in $n=2$ and $n=4$ show that GINN-based detectors achieve high true-positive rates and robust generalization, outperforming MLP-based counterparts and enabling edge-detection tasks with reduced function evaluations. The approach yields portable, reusable detectors that can be integrated into various numerical algorithms, with public code availability and clear practical considerations, though synthetic data generation remains a notable upfront cost to scale to higher dimensions.

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.

Figures (13)

• Figure 1: Example of sparse grid in $\mathbb{R}^2$ obtained with $[\alpha_i,\beta_i]=[-1,1]$ for $i=1,2$, level-to-knot function $m$ (see \ref{['eq:ltk_func_doubling']}), equispaced knots $\mathcal{K}^{[2]}_{i,h}$, and multi-indices set $\mathcal{I}_{\text{sum}}(6)$. Black dotted lines depict the boundary of the sparse grid box (see \ref{['def:sg_box']}).
• Figure 2: Building a sparse grid graph based on the sparse grid $\mathcal{S}$ in \ref{['fig:sg_cross_2d_noedges']}. Left. Raw graph obtained connecting the aligned and consecutive points of the sparse grid. Right. Final graph obtained removing from the raw one the edges that intersect an edge of length equal or shorter. The edge opacity depends on the edge weights (see \ref{['def:wgraph_sg']}).
• Figure 3: Example of sparse grid points classification made for a function $g:\mathbb{R}^2\to\mathbb{R}$ with an exact discontinuity detector $\Delta^*$ (left) and a inexact discontinuity detector $\Delta$ (right). The black dotted line is the function's discontinuity interface. Green circles are true troubled points, magenta circles are false troubled points, magenta dots are false non-troubled points. For the inexact detector, we report also the values $p_i$ corresponding to the highlighted sparse grid points (threshold $\tau=0.5$).
• Figure 4: Example of three steps of \ref{['alg:sg_alg_detection']}, assuming an exact detector $\Delta^*$ and a SGG as in \ref{['fig:graph_from_sg_firstexample']}-right. Opaque green circles are the troubled points identified with the current grid. Transparent green circles are the troubled points identified at the previous steps and where the next sparse grids will be centered.
• Figure 5: Visual representation of \ref{['eq:ginn_node_action']}. Example with $n=4$ nodes (non-directed graph), $i=1$; for simplicity, the bias is not illustrated.

Theorems & Definitions (22)

• Definition 2.1: Sparse Grid Box
• Definition 2.2: Sparse Grid Similarity
• Definition 2.3: Sparse Grid Graph
• Definition 2.4: Weighted Sparse Grid Graph
• Proposition 2.1
• proof
• Definition 2.5: Sparse Grid Graph Similarity
• Remark 2.1: Weights for graphs based on equispaced sparse grids with hypercubic box
• Definition 3.1: Troubled Points
• Definition 3.2: Discontinuity Detector