Learning Mesh-Free Discrete Differential Operators with Self-Supervised Graph Neural Networks
This work addresses a domain-specific problem in computational physics and engineering by providing a more efficient and accurate method for mesh-free simulations, though it is incremental as it builds on existing neural network and mesh-free techniques.
The paper tackles the problem of mesh-free discrete differential operators, which often sacrifice accuracy for computational cost or vice versa, by introducing a self-supervised graph neural network framework that learns these operators from local geometry, achieving improved accuracy over Smoothed Particle Hydrodynamics and a favorable accuracy-cost trade-off in moderate-accuracy regimes.
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.