# Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems

arXiv: https://arxiv.org/abs/2402.11722 · HTML: https://mlsanity.com/paper/2402.11722/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 40 references that are in the map.
- First posted on arXiv 18 Feb 2024.

## Builds on (most cited first, 26 of 26)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Learning to Solve PDE-constrained Inverse Problems with Graph Networks](https://mlsanity.com/paper/2206.00711/) (2022)
- [Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems](https://mlsanity.com/paper/2209.02772/) (2022)
- [Multi-Resolution Active Learning of Fourier Neural Operators](https://mlsanity.com/paper/2309.16971/) (2023)
- [DeepOHeat: Operator Learning-based Ultra-fast Thermal Simulation in 3D-IC Design](https://mlsanity.com/paper/2302.12949/) (2023)
- [A Kernel Approach for PDE Discovery and Operator Learning](https://mlsanity.com/paper/2210.08140/) (2022)
- [Parameterizations for Ensemble Kalman Inversion](https://mlsanity.com/paper/1709.01781/) (2017)
- [A Bayesian Level Set Method for Geometric Inverse Problems](https://mlsanity.com/paper/1504.00313/) (2015)

## Built on by (most cited first, 19 of 19)

- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Models](https://mlsanity.com/paper/2506.13754/) (2025)
- [FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning](https://mlsanity.com/paper/2509.12344/) (2025)
- [Learning Hidden Physics and System Parameters with Deep Operator Networks](https://mlsanity.com/paper/2412.05133/) (2024)
- [Diffusion models as probabilistic neural operators for recovering unobserved states of dynamical systems](https://mlsanity.com/paper/2405.07097/) (2024)
- [FNOPE: Simulation-based inference on function spaces with Fourier Neural Operators](https://mlsanity.com/paper/2505.22573/) (2025)
- [A unified physics-informed generative operator framework for general inverse problems](https://mlsanity.com/paper/2511.03241/) (2025)
- [ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidance](https://mlsanity.com/paper/2601.20227/) (2026)
- [Inverse Neural Operator for ODE Parameter Optimization](https://mlsanity.com/paper/2603.11854/) (2026)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026)
- [GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems](https://mlsanity.com/paper/2607.01128/) (2026)
- [Neural Persistence Dynamics](https://mlsanity.com/paper/2405.15732/) (2024)
- [Deep Optimal Sensor Placement for Black Box Stochastic Simulations](https://mlsanity.com/paper/2410.12036/) (2024)
- [Deep Generative Prior for First Order Inverse Optimization](https://mlsanity.com/paper/2504.20278/) (2025)
- [Estimating Parameter Fields in Multi-Physics PDEs from Scarce Measurements](https://mlsanity.com/paper/2509.00203/) (2025)
- [Learning Generalizable Neural Operators for Inverse Problems](https://mlsanity.com/paper/2512.18120/) (2025)
- [DiffUNet^2: Bidirectional Conditional Diffusion for Probabilistic Scientific Spatiotemporal Modeling](https://mlsanity.com/paper/2606.03926/) (2026)
- [SBNO : Schr"odinger Bridge Neural Operator for the Forward and Inverse Problems under Missing Information](https://mlsanity.com/paper/2610.13317/) (2026)

## Abstract (arXiv)

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.