Not All Interactions Need to Be Global: Query-Centered Multi-Scale Spatial Observation for Neural Operators

Neural operators provide resolution-independent surrogates for partial differential equations, but accurately modeling spatial dependencies across different ranges remains challenging. While local structures and broad spatial configurations jointly determine the solution, existing operators typically rely on global representations or extensive spatial interactions to acquire long-range context. We propose theMulti-Scale Spatial Observation Operator (MSO), which reformulates long-range dependency modeling as a query-centered observation problem. MSO constructs a hierarchical spatial representation and propagates information independently along high-to-low and low-to-high resolution paths. For each query coordinate, it extracts spatially aligned features from all resolution levels and propagation directions, and adaptively aggregates these $2(L+1)$ observations through scale-wise attention. This design enables broad spatial observation while maintaining sparse query-side interaction, avoiding dense query-to-token mixing. Experiments on diverse PDE benchmarks covering incompressible and compressible fluid dynamics, shallow-water flows, and diffusion-reaction dynamics demonstrate strong predictive performance with a substantially smaller parameter footprint than representative neural operators. Further analyses show that different query locations favor different observation scales, supporting the effectiveness of query-adaptive multi-scale spatial observation.