Learning Event-to-Field Operators Without Interpolation

A point source in a PDE does not occupy an entire grid. A weather station reports one value at one site; it does not observe the temperature between stations. A pickup event in a city occurs at a coordinate and a time; it is not born as an image. Yet many neural-operator pipelines begin by turning such sparse observations into dense surrogate fields before learning. This interpolation step is often treated as harmless preprocessing, but in sparse regimes it changes the object on which the operator is asked to act. We introduce the Dirac Neural Operator (DIRAC-NO), a measure-native neural operator for sparse event-to-field learning without interpolation. DIRAC-NO keeps observations as atomic event measures until query time and constructs the output field through adaptive Dirac-kernel aggregation. Because this aggregation is implemented by direct summation, the front end preserves source additivity at the aggregation stage and admits a learned Green-kernel view of source-to-field response. Across controlled point-source PDE benchmarks and real sparse reconstruction tasks, DIRAC-NO exposes the cost of interpolation-first learning. On MeasureBench-1D, it achieves relative $L^2$ error 0.015 at $n=128$, compared with 0.49 for FNO, and it better preserves superposition structure in trained models. In 2D, it is strongest under parameter and density shift, while ERA5 reconstruction shows that the representation advantage persists beyond analytic PDEs at high observation budgets. Boundary cases and negative-control event streams further show that the benefit appears where the representation argument predicts: sparse, event-native tasks with source-response structure. These results suggest that sparse operator learning should be organized not only by the neural backbone, but by the native space in which the input is represented.