Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models
This provides a method for solving stochastic partial differential equations in various settings, but it appears incremental as it builds on existing normalizing flow techniques.
The authors tackled the problem of learning random fields from scattered measurements by introducing normalizing field flows (NFF), which use a bijective transformation between Gaussian and target stochastic fields, achieving a unified framework for solving forward, inverse, and mixed stochastic partial differential equations.
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.