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.

Partial differential equations (PDEs) play a dominant role in the mathematical modeling of many complex dynamical processes. Solving these PDEs often requires prohibitively high computational costs, especially when multiple evaluations must be made for different parameters or conditions. After training, neural operators can provide PDEs solutions significantly faster than traditional PDE solvers. In this work, invariance properties and computational complexity of two neural operators are examined for transport PDE of a scalar quantity. Neural operator based on graph kernel network (GKN) operates on graph-structured data to incorporate nonlocal dependencies. Here we propose a modified formulation of GKN to achieve frame invariance. Vector cloud neural network (VCNN) is an alternate neural operator with embedded frame invariance which operates on point cloud data. GKN-based neural operator demonstrates slightly better predictive performance compared to VCNN. However, GKN requires an excessively high computational cost that increases quadratically with the increasing number of discretized objects as compared to a linear increase for VCNN.