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

TL;DR

This paper introduces normalizing field flows, a neural network-based method for learning and solving stochastic partial differential equations by transforming Gaussian fields into target stochastic fields, applicable to forward, inverse, and mixed problems.

Contribution

The paper presents a novel normalizing flow framework for stochastic fields that can handle non-Gaussian processes and solve various stochastic PDEs in a unified manner.

Findings

Successfully learns non-Gaussian stochastic processes.

Effectively solves forward and inverse stochastic PDEs.

Demonstrates versatility across different stochastic models.

Abstract

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.