Learning Mesh-Free Discrete Differential Operators with Self-Supervised Graph Neural Networks

TL;DR

This paper presents a graph neural network framework that learns mesh-free discrete differential operators, achieving improved accuracy and robustness for complex geometries, and demonstrating applicability to fluid dynamics simulations.

Contribution

Introduces a novel GNN-based approach for learning mesh-free differential operators that are accurate, geometry-robust, and reusable across different configurations.

Findings

Learned operators match classical polynomial consistency.

Achieve better accuracy than SPH methods.

Show favorable accuracy-cost trade-offs in fluid simulations.

Abstract

Mesh-free numerical methods provide flexible discretisations for complex geometries; however, classical meshless discrete differential operators typically trade low computational cost for limited accuracy or high accuracy for substantial per-stencil computation. We introduce a parametrised framework for learning mesh-free discrete differential operators using a graph neural network trained via polynomial moment constraints derived from truncated Taylor expansions. The model maps local stencils relative positions directly to discrete operator weights. The current work demonstrates that neural networks can learn classical polynomial consistency while retaining robustness to irregular neighbourhood geometry. The learned operators depend only on local geometry, are resolution-agnostic, and can be reused across particle configurations and governing equations. We evaluate the framework using standard numerical analysis diagnostics, showing improved accuracy over Smoothed Particle Hydrodynamics, and a favourable accuracy-cost trade-off relative to a representative high-order consistent mesh-free method in the moderate-accuracy regime. Applicability is demonstrated by solving the weakly compressible Navier-Stokes equations using the learned operators.