Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems

TL;DR

iFNO tackles forward and inverse PDE problems within a single, invertible framework by stacking invertible Fourier blocks in a latent space and coupling them with a beta-VAE for posterior sampling. This design enables efficient bidirectional learning through shared parameters and provides uncertainty estimates for inverse predictions. Across seven benchmark PDE tasks, iFNO delivers state-of-the-art or near-best predictive performance and demonstrates meaningful uncertainty calibration, particularly in regions with interfaces or boundaries. The approach offers a principled, scalable path for robust joint forward/inverse inference and can be extended to other neural-operator architectures.

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.

Figures (12)

• Figure 1: An Overview of iFNO. Left panel: the $\beta$-VAE module and the invertible Fourier blocks. Right panel: the entire architecture. "SP" and "RSP" denote softplus and the reciprocal of the softplus, respectively. We first train separately the $\beta$-VAE module and invertible Fourier blocks as shown in the first panel. Then we combine them to continue training the entire model as depicted in the right panel.
• Figure 2: iFNO Pointwise Inverse Prediction Error and Predictive Standard Deviation, denoted by "Error" and "SD" respectively followed with noise levels in training data.
• Figure 3: An Illustration of The Seismic Surveys, where the shaded region is the physical media of interest and dashed lines are the interface for different regions of square slowness.
• Figure 4: Pointwise Error of Inverse Prediction, where "-0%" and "-10%" indicate the noise level in the training data.
• Figure 5: Pointwise Error of Forward Prediction, where "-0%" and "-10%" indicate the noise level in the training data.