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Problem
Problem
Many physical quantities, such as temperature, humidity or a pollutant's concentration, are continuous fields over a city, but they are measured at a few dozen fixed points. Models are usually asked to reconstruct the whole field from those points. When the network misses the sites that drive the local dynamics, however, two different fields can produce the same readings at every sensor, and no method can tell them apart.
FieldFormer predicts at a query point from a local neighbourhood of sensors, encoded with learned scales and passed to a transformer encoder. From the TMLR 2026 paper.
Approach and findings
Approach and findings
FieldFormer makes that limit precise and builds a model around it. We proved that if sensors observe a random fraction of the sites, any estimator's error on one of two indistinguishable completions is bounded below in proportion to the fraction of sites left unobserved. The model therefore predicts on the support that the deployed network observes. For each query it attends to a fixed number of nearby sensors and recent times, and it learns how far each direction of space and time should count as near, so the cost of a prediction does not grow with the size of the dataset.
On filling in missing readings at Delhi's sensors, FieldFormer's error (RMSE) was 22.06 µg/m³ for PM2.5 and 39.25 µg/m³ for PM10, against 32.91 and 53.84 for RecFNO, a neural-operator baseline. On reconstructing the full field and on predicting at held-out sensors, no method dominates, which is what the lower bound predicts.
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