Normalizing Field Flows: Solving forward and inverse stochastic differential equations using Physics-Informed flow model

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 reference random field (say, a Gaussian random field with the Karhunen-Lo\`eve 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, mixed Gaussian processes, and forward & inverse stochastic partial differential equations.