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Meshless method stencil evaluation with machine learning

Miha Rot, Aleksandra Rashkovska

TL;DR

This work addresses the challenge of selecting high-quality stencils in local meshless PDE solvers by leveraging a labelled stencil dataset generated via RBF-FD with a polyharmonic $r^3$ RBF and $2^{nd}$-order augmentation. A modified PointNet classifier is trained to predict stencil quality across multiple sizes, enabling cross-size evaluation without solving the weight system upfront. Results show strong discrimination for extreme stencil classes, with AUCs around $0.89$–$0.94$ and a best-case scenario where about 97% of $Q_1$ stencils lie below the median error $\epsilon$ and nearly all $Q_4$ lie above it (median $\epsilon = 3.9 \cdot 10^{-2}$), highlighting the method's potential for practical stencil construction. The study suggests that padding across sizes allows a single model to support multiple stencil configurations, though further improvements in accuracy and explainability are needed for deployment.

Abstract

Meshless methods are an active and modern branch of numerical analysis with many intriguing benefits. One of the main open research questions related to local meshless methods is how to select the best possible stencil - a collection of neighbouring nodes - to base the calculation on. In this paper, we describe the procedure for generating a labelled stencil dataset and use a variation of pointNet - a deep learning network based on point clouds - to create a classifier for the quality of the stencil. We exploit features of pointNet to implement a model that can be used to classify differently sized stencils and compare it against models dedicated to a single stencil size. The model is particularly good at detecting the best and the worst stencils with a respectable area under the curve (AUC) metric of around 0.90. There is much potential for further improvement and direct application in the meshless domain.

Meshless method stencil evaluation with machine learning

TL;DR

This work addresses the challenge of selecting high-quality stencils in local meshless PDE solvers by leveraging a labelled stencil dataset generated via RBF-FD with a polyharmonic RBF and -order augmentation. A modified PointNet classifier is trained to predict stencil quality across multiple sizes, enabling cross-size evaluation without solving the weight system upfront. Results show strong discrimination for extreme stencil classes, with AUCs around – and a best-case scenario where about 97% of stencils lie below the median error and nearly all lie above it (median ), highlighting the method's potential for practical stencil construction. The study suggests that padding across sizes allows a single model to support multiple stencil configurations, though further improvements in accuracy and explainability are needed for deployment.

Abstract

Meshless methods are an active and modern branch of numerical analysis with many intriguing benefits. One of the main open research questions related to local meshless methods is how to select the best possible stencil - a collection of neighbouring nodes - to base the calculation on. In this paper, we describe the procedure for generating a labelled stencil dataset and use a variation of pointNet - a deep learning network based on point clouds - to create a classifier for the quality of the stencil. We exploit features of pointNet to implement a model that can be used to classify differently sized stencils and compare it against models dedicated to a single stencil size. The model is particularly good at detecting the best and the worst stencils with a respectable area under the curve (AUC) metric of around 0.90. There is much potential for further improvement and direct application in the meshless domain.
Paper Structure (8 sections, 1 equation, 7 figures, 1 table)

This paper contains 8 sections, 1 equation, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Data preparation
  3. Stencil generation
  4. Test fields
  5. Method
  6. Results
  7. Future work
  8. Conclusions

Figures (7)

  • Figure 1: The best 3 stencils with $s = 15$ in the top row and the worst 3 in the bottom. The red point marks the central node where we approximate the operator. Positions are centred and normalised.
  • Figure 2: Distribution of error measure $\epsilon$ in datasets for stencils with different sizes $s$. The smaller stencils have a much higher variation.
  • Figure 3: Modified PointNet neural network architecture for $s=15$. Visualisation created directly from the Keras model using the Net2Vis Net2Vis framework.
  • Figure 4: Decrease in loss function and increase in accuracy during the training for three individual stencil sizes and the mix.
  • Figure 5: Confusion matrix with normalized columns for the mixed stencil size model with $s \in \{6, 7, 9, 12, 15\}$. Values on the diagonal are the correctly identified classes.
  • ...and 2 more figures