Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems

TL;DR

This paper introduces an invertible Fourier Neural Operator (iFNO) that jointly addresses forward and inverse problems, leveraging invertible blocks and a variational auto-encoder to improve performance on ill-posed inverse tasks.

Contribution

The paper proposes a novel invertible Fourier Neural Operator with shared parameters and integrated VAE for bi-directional problem solving, enhancing inverse problem handling.

Findings

Demonstrated superior performance on seven benchmark tasks.

Effectively mitigated ill-posedness and data scarcity issues.

Enabled joint forward and inverse problem solving with a unified model.

Abstract

Fourier Neural Operator (FNO) is a powerful and popular operator learning method. However, FNO is mainly used in forward prediction, yet a great many applications rely on solving inverse problems. In this paper, we propose an invertible Fourier Neural Operator (iFNO) for jointly tackling the forward and inverse problems. We developed a series of invertible Fourier blocks in the latent channel space to share the model parameters, exchange the information, and mutually regularize the learning for the bi-directional tasks. We integrated a variational auto-encoder to capture the intrinsic structures within the input space and to enable posterior inference so as to mitigate challenges of illposedness, data shortage, noises that are common in inverse problems. We proposed a three-step process to combine the invertible blocks and the VAE component for effective training. The evaluations on seven benchmark forward and inverse tasks have demonstrated the advantages of our approach.