Green-Routed Neural Operators: Physics Determines Where the Network Reads

We identify a mismatch between the physical role of transport fields in many PDEs and their usual role in neural operators: PDEs use them to select read coordinates, whereas neural operators typically treat them only as input values. We address this mismatch with the Green-Routed Neural Operator (GRNO), which uses the governing equation to determine where latent features are sampled. A parameter-free equation adapter evaluates the diagnostic relation and constructs a departure map whose values are the read coordinates. A multiscale encoder-decoder combines centered and routed reads of latent features to learn the complete finite-time update. Across five two- and three-dimensional PDE systems, GRNO achieves the lowest mean final relative $L^2$ error on four under 40-step autoregressive evaluation and remains competitive on Keller–Segel. Fixed-weight route interventions reveal strong dependence on direction and spatial alignment in four systems, with weak dependence in Keller–Segel. In independently trained ablations, GRNO achieves lower mean errors than variants that supply the transport field only as an input feature, substitute a learned displacement for the equation-specified route, or apply the route with a spatial misalignment, across all five systems. It also substantially outperforms directly advecting the physical state and learning the remaining update, indicating that equation-specified read coordinates provide an effective structural prior for long-horizon PDE forecasting.