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-Loève (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.

　　Publication: 

　　Journal of Computational Physics, Volume 461, Issue C, Jul 2022. 

　　Author: 

　　Ling Guo 

　　Department of Mathematics, Shanghai Normal University, Shanghai, China 

　　Hao Wu 

　　School of Mathematical Sciences, Tongji University, Shanghai, China 

　　Tao Zhou 

　　LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China. 

　　E-mail: tzhou@lsec.cc.ac.cn