mlsanityData to 7 Oct 2026

Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems

arXiv 2402.11722 ↗First posted 18 Feb 2024cs.LGcited by 18

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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.

Key facts. Cited by 18 papers in the map (data through 7 Oct 2026). Builds on 26 of its 40 references that are in the map. First posted on arXiv 18 Feb 2024.

Builds on

26 of 40 references.

References · 26

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  2. 2013Auto-Encoding Variational Bayescited by 3,980
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  4. 2016Density estimation using Real NVPcited by 2,029
  5. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  6. 2020Learning Mesh-Based Simulation with Graph Networkscited by 840
  7. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
  8. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  9. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  10. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 396
  11. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  12. 2021Choose a Transformer: Fourier or Galerkincited by 333
14 more
  1. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 321
  2. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
  3. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  4. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
  5. 2022NOMAD: Nonlinear Manifold Decoders for Operator Learningcited by 81
  6. 2023Neural Inverse Operators for Solving PDE Inverse Problemscited by 61
  7. 2023Kernel Methods are Competitive for Operator Learningcited by 50
  8. 2022Learning to Solve PDE-constrained Inverse Problems with Graph Networkscited by 37
  9. 2022Semi-supervised Invertible Neural Operators for Bayesian Inverse Problemscited by 25
  10. 2023Multi-Resolution Active Learning of Fourier Neural Operatorscited by 22
  11. 2023DeepOHeat: Operator Learning-based Ultra-fast Thermal Simulation in 3D-IC Designcited by 13
  12. 2022A Kernel Approach for PDE Discovery and Operator Learningcited by 7
  13. 2017Parameterizations for Ensemble Kalman Inversioncited by 6
  14. 2015A Bayesian Level Set Method for Geometric Inverse Problemscited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 5; 2025: 8; 2026: 5.

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  1. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 100
  2. 2025Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEscited by 9
  3. 2025VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Modelscited by 7
  4. 2025FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learningcited by 7
  5. 2024Learning Hidden Physics and System Parameters with Deep Operator Networkscited by 6
  6. 2024Diffusion models as probabilistic neural operators for recovering unobserved states of dynamical systemscited by 4
  7. 2025FNOPE: Simulation-based inference on function spaces with Fourier Neural Operatorscited by 3
  8. 2025A unified physics-informed generative operator framework for general inverse problemscited by 1
  9. 2026ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidancecited by 1
  10. 2026Inverse Neural Operator for ODE Parameter Optimizationcited by 1
  11. 2024Neural Persistence Dynamicscited by 0
  12. 2024Deep Optimal Sensor Placement for Black Box Stochastic Simulationscited by 0
6 more
  1. 2025Deep Generative Prior for First Order Inverse Optimizationcited by 0
  2. 2025Estimating Parameter Fields in Multi-Physics PDEs from Scarce Measurementscited by 0
  3. 2025Learning Generalizable Neural Operators for Inverse Problemscited by 0
  4. 2026Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospectscited by 0
  5. 2026DiffUNet^2: Bidirectional Conditional Diffusion for Probabilistic Scientific Spatiotemporal Modelingcited by 0
  6. 2026GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problemscited by 0