Executable Neural Operators: Routing Local Physical Mechanisms in Continuous Spatial Dynamics

While Neural Operators and Physics-Informed Neural Networks (PINNs) have accelerated the simulation of continuous spatial data, they predominantly rely on global implicit mappings. This monolithic representation can entangle local physical rules, rendering models brittle to non-stationary mutations in complex media, such as sudden changes in boundary conditions or unobserved subsurface scatterers. To address this limitation, we propose the Executable Neural Operator (ENO), a novel architecture that represents localized physical mechanisms within continuous fields. By leveraging a unified bipartite graph embedding, ENO tokenizes large-scale continuous spatial data into a routed graph. Instead of global spectral mixing, ENO employs Dynamic Operator Routing (DOR) to assign localized neural programs to simulate wave propagation and energy transfer. During medium property mutations, our Targeted Mechanism Revision (TMR) protocol provides gradient isolation under explicit locality and parameter-ownership conditions. We extensively evaluate ENO on high-contrast 3D magnetotelluric dynamics. Results demonstrate that ENO achieves competitive zero-shot accuracy on unseen out-of-distribution geometries while suppressing the physics violation rate and reducing background prediction drift, establishing a theoretical and empirical basis for mechanism discovery in non-stationary physical systems.