[2603.24641] Learning Mesh-Free Discrete Differential Operators with Self-Supervised Graph Neural Networks

Computer Science > Machine Learning arXiv:2603.24641 (cs) [Submitted on 25 Mar 2026] Title:Learning Mesh-Free Discrete Differential Operators with Self-Supervised Graph Neural Networks Authors:Lucas Gerken Starepravo, Georgios Fourtakas, Steven Lind, Ajay B. Harish, Tianning Tang, Jack R. C. King View a PDF of the paper titled Learning Mesh-Free Discrete Differential Operators with Self-Supervised Graph Neural Networks, by Lucas Gerken Starepravo and 5 other authors View PDF HTML (experimental) 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...