Learning mesh-free discrete differential operators with self-supervised graph neural networks

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 neural networks 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, and can be reused across particle configurations and governing equations. The framework introduces an additional design axis absent from traditional discretisations: the learned operators have a capacity-dependent error floor, so a model may be optimised for accuracy, for computational cost, or a balance of the two. A trained model may be applied across resolutions, though accuracy cannot be refined below the limiting error set by model capacity. We evaluate the framework using standard numerical analysis diagnostics, showing improved accuracy over Smoothed Particle Hydrodynamics, and applicability is demonstrated by solving the Poisson’s equation and the weakly compressible Navier–Stokes equations using the learned operators.