Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models (Ling Guo, Hao Wu, Tao Zhou)


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.



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



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.