  We introduce in this work the normalizing field flows (NFF) for learning
random fields from scattered measurements. More precisely, we construct a
bijective transformation (a normalizing flow characterizing by neural networks)
between a Gaussian random field with the Karhunen-Lo\`eve (KL) expansion
structure and the target stochastic field, where the KL expansion coefficients
and the invertible networks are trained by maximizing the sum of the
log-likelihood on scattered measurements. This NFF model can be used to solve
data-driven forward, inverse, and mixed forward/inverse stochastic partial
differential equations in a unified framework. We demonstrate the capability of
the proposed NFF model for learning Non Gaussian processes and different types
of stochastic partial differential equations.
