  Motivated by objects such as electric fields or fluid streams, we study the
problem of learning stochastic fields, i.e. stochastic processes whose samples
are fields like those occurring in physics and engineering. Considering general
transformations such as rotations and reflections, we show that spatial
invariance of stochastic fields requires an inference model to be equivariant.
Leveraging recent advances from the equivariance literature, we study
equivariance in two classes of models. Firstly, we fully characterise
equivariant Gaussian processes. Secondly, we introduce Steerable Conditional
Neural Processes (SteerCNPs), a new, fully equivariant member of the Neural
Process family. In experiments with Gaussian process vector fields, images, and
real-world weather data, we observe that SteerCNPs significantly improve the
performance of previous models and equivariance leads to improvements in
transfer learning tasks.
