Artificial neural network evaluation of geometric constants for polygonal domains

TL;DR

The paper presents a data-driven framework that uses feed-forward neural networks to estimate geometry-dependent constants ($C_p$, $C_t$, $C_i$) arising in Poincaré, trace, and inverse inequalities for polygonal domains. By linking a compact set of geometric quality metrics to the constants through supervised regression, the approach replaces expensive eigenvalue computations with fast online predictions once offline training is complete. The method demonstrates high-accuracy predictions on both non-convex and convex polygons and proves adaptable to polygons of arbitrary size, with a scalable rescaling strategy. The authors emphasize offline data generation, shallow yet effective network architectures, and the potential to extend the framework to 3D polytopes and additional geometric constants, offering a practical tool for PEM design and PDE analysis.

Abstract

We propose an approach based on Artificial Neural Networks (ANNs) to evaluate geometric constants relevant to the analysis and design of numerical schemes for partial differential equations. These constants play a central role, significantly influencing, for instance, a posteriori error estimates and the overall design of the computational strategy. Our technique leverages ANNs to learn the dependencies between these constants and a set of descriptive geometric features associated to polytopal mesh elements. The main computational costs are confined to data processing and training phases, which can be performed offline once and for all. This yields an effective tool for computing the constants, which we verify and show to be applicable across different scenarios, without substantial modifications - demonstrating its broader usability beyond the specific example considered.

Figures (12)

• Figure 1: Example of non-convex polygons (a); a Voronoi mesh and convex polygon extracted from the mesh (b).
• Figure 2: Overview of the proposed approach: the geometric metrics are computed for a polygon $K$ to obtain the evaluation of the considered constant corresponding to $K$.
• Figure 3: Boxplots of the attributes in the non-convex (top) and convex (bottom) training sets, before (left) and after (right) outliers elimination.
• Figure 4: Non-convex polygons -- Poincaré ineqiality. Validation loss evaluated at each of the first $1500$ training epochs for various values of network depth $L \in \{1, 3, 5, 7\}$, width $N \in \{8, 32, 64, 128, 256, 512, 1024\}$, and learning rate $\eta \in \{10^{-2},\ 5{\cdot}10^{-3},\ 10^{-3},\ 5{\cdot}10^{-4},\ 10^{-4}\}$.
• Figure 5: (a) Statistical analysis of the training target labels for learning the Poincaré constant on non-convex polygons: the data (blue), the mean value (read), the standard deviation (green and olive), the minimum (cyan) and the maximum values (magenta). (b) Training and validation loss over epochs for Poincaré inequality on non-convex polygons.