Mesh-free numerical methods have significant potential, as discretisation relies solely on local connectivity information between collocation points without requiring topological connectivity. Developed in the 1970s, Smoothed Particle Hydrodynamics (SPH) approximates differential operators via radial kernel interpolation over neighbouring particles. While this approach is computationally efficient, it is inherently inconsistent, significantly limiting the accuracy and stability that SPH can achieve. This motivated the development of more accurate numerical methods that enforce polynomial consistency through the solution of a local linear system. However, solving linear systems introduces significant computational head, which can be prohibitely expensive in certain cases, such as in Lagrangian simulations. In this work, we introduce Neural Mesh-Free Differential Operators (NeMDO), a physics-agnostic framework for learning discrete mesh-free differential operators directly from irregular particle configurations using a self-supervised graph neural network. Rather than learning PDE solutions, closures, or problem-specific dynamics, NeMDO learns local operator weights solely from geometric information by enforcing polynomial consistency constraints derived from a truncated Taylor expansion. A single trained network is reused across all particle stencils, enabling resolution- and configuration-agnostic operator construction without retraining. The learned operators can be directly substituted for classical mesh-free operators within existing solvers, requiring no modification to the code structure. We show that graph neural networks can accurately learn these consistency constraints and predict operator weights exhibiting structural properties analogous to those of classical consistent mesh-free discretisations. The learned operators are rigorously validated using established numerical analysis tools, including convergence studies, modal response analysis, stability assessments, and ablation studies. Finally, we demonstrate applicability by solving the weakly compressible Navier–Stokes equations. Results show improved accuracy over SPH and a favourable accuracy–cost trade-off relative to a representative high-order consistent mesh-free method in the moderate-accuracy regime.