mlsanityData to 7 Oct 2026

Fourier Neural Operator for Parametric Partial Differential Equations

arXiv 2010.08895 ↗First posted 18 Oct 2020cs.LG cs.NA math.NAcited by 2,806

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Cited by 139 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.

Key facts. Cited by 2,806 papers in the map (data through 7 Oct 2026). First posted on arXiv 18 Oct 2020.

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Citing papers per year: 2021: 112; 2022: 240; 2023: 366; 2024: 517; 2025: 703; 2026: 868.

2,806 papers.

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  1. 2022FEDformer: Frequency Enhanced Decomposed Transformer for Long-term Series Forecastingcited by 1,200
  2. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
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  4. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  5. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  6. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
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  10. 2022Message Passing Neural PDE Solverscited by 368
  11. 2021FNet: Mixing Tokens with Fourier Transformscited by 343
  12. 2021Choose a Transformer: Fourier or Galerkincited by 333
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  4. 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 293
  5. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  6. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
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  10. 2021Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformerscited by 237
  11. 2021Factorized Fourier Neural Operatorscited by 237
  12. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  13. 2023Spherical Fourier Neural Operators: Learning Stable Dynamics on the Spherecited by 225
  14. 2022U-NO: U-shaped Neural Operatorscited by 219
  15. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  16. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
  17. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 205
  18. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
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  20. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  21. 2022Fast Sampling of Diffusion Models via Operator Learningcited by 173
  22. 2022Forecasting Global Weather with Graph Neural Networkscited by 165
  23. 2021Next Generation Reservoir Computingcited by 160
  24. 2022FiLM: Frequency improved Legendre Memory Model for Long-term Time Series Forecastingcited by 147
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  27. 2024The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learningcited by 134
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  31. 2022Physics-Informed Deep Neural Operator Networkscited by 110
  32. 2023Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictorscited by 110
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  34. 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 103
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  38. 2021Neural Flows: Efficient Alternative to Neural ODEscited by 99

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