  Fourier Neural Operator (FNO) is a popular operator learning method, which
has demonstrated state-of-the-art performance across many tasks. However, FNO
is mainly used in forward prediction, yet a large family of applications rely
on solving inverse problems. In this paper, we propose an invertible Fourier
Neural Operator (iFNO) that tackles both the forward and inverse problems. We
designed a series of invertible Fourier blocks in the latent channel space to
share the model parameters, efficiently 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 overcome challenges of
illposedness, data shortage, noises, etc. We developed a three-step process for
pre-training and fine tuning for efficient training. The evaluations on five
benchmark problems have demonstrated the effectiveness of our approach.
