AURA: Graph Learning via Superposed Neural Fields Beyond Message Passing

Message-passing graph neural networks update each node from a fixed-radius neighbourhood, so a depth-$L$ model cannot use information more than $L$ hops away. Many relational systems, molecules among them, instead interact through fields that superpose contributions from distributed sources. We introduce AURA, a graph layer built on source-medium-response computation: nodes emit source coefficients, a learned sparse medium operator propagates them through a truncated resolvent solve, and each node updates by probing the value, gradient, and curvature of the resulting field. Local linear message passing is a degenerate case of AURA, an exact solve aggregates walks of every length, and for every depth $L$ we construct tasks on which all depth-$L$ local message-passing networks are at chance while a constant-depth AURA model is exact. We test the mechanism with controls. On three interference tasks, local baselines stay at chance, full-attention graph transformers solve every task (a positive control), and AURA reaches 79–90% mean accuracy; replacing only AURA's learned medium with a uniform one, at identical depth, receptive field, and parameter count, returns all three tasks to chance. On a shared QM9 protocol and a QM9-derived field-reconstruction diagnostic, which bound the scope of our claims, AURA has the lowest mean error among the evaluated models, at 1.6$\times$ the per-run wall-clock cost of a GPS-style transformer on QM9. Code and all per-run result files are provided as supplementary material.