Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models

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.