🤖 AI Summary
This work addresses the longstanding challenge in meshfree methods of simultaneously achieving high accuracy, computational efficiency, and robustness to irregular point distributions when approximating differential operators. The authors propose a self-supervised graph neural network that directly learns discrete differential operator weights from local point-set geometry, incorporating polynomial moment constraints derived from truncated Taylor expansions to enforce consistency. This approach yields, for the first time, a high-accuracy operator that depends solely on local geometry, is resolution-independent, and generalizes across diverse point configurations—combining the classical polynomial reproduction property with robustness to arbitrary node layouts. Experiments demonstrate superior accuracy over standard SPH on benchmark problems, a more favorable accuracy–efficiency trade-off than high-order consistent meshfree methods at moderate precision levels, and successful application to solving weakly compressible Navier–Stokes equations.
📝 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.