mlsanityData to 7 Oct 2026

Factorized Fourier Neural Operators

arXiv 2111.13802 ↗First posted 27 Nov 2021cs.LG cs.CEcited by 237

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Abstract (arXiv)

We propose the Factorized Fourier Neural Operator (F-FNO), a learning-based approach for simulating partial differential equations (PDEs). Starting from a recently proposed Fourier representation of flow fields, the F-FNO bridges the performance gap between pure machine learning approaches to that of the best numerical or hybrid solvers. This is achieved with new representations - separable spectral layers and improved residual connections - and a combination of training strategies such as the Markov assumption, Gaussian noise, and cosine learning rate decay. On several challenging benchmark PDEs on regular grids, structured meshes, and point clouds, the F-FNO can scale to deeper networks and outperform both the FNO and the geo-FNO, reducing the error by 83% on the Navier-Stokes problem, 31% on the elasticity problem, 57% on the airfoil flow problem, and 60% on the plastic forging problem. Compared to the state-of-the-art pseudo-spectral method, the F-FNO can take a step size that is an order of magnitude larger in time and achieve an order of magnitude speedup to produce the same solution quality.

Key facts. Cited by 237 papers in the map (data through 7 Oct 2026). Builds on 17 of its 33 references that are in the map. First posted on arXiv 27 Nov 2021.

Builds on

17 of 33 references.

References · 17

  1. 2017Attention Is All You Needcited by 59,932
  2. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  3. 2016SGDR: Stochastic Gradient Descent with Warm Restartscited by 4,993
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  5. 2020Learning to Simulate Complex Physics with Graph Networkscited by 867
  6. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  7. 2021Machine learning accelerated computational fluid dynamicscited by 461
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  9. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  10. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  11. 2021FNet: Mixing Tokens with Fourier Transformscited by 343
  12. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
5 more
  1. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 203
  2. 2020Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solverscited by 201
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 8; 2023: 32; 2024: 54; 2025: 70; 2026: 73.

237 papers.

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  1. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 321
  2. 2024Transolver: A Fast Transformer Solver for PDEs on General Geometriescited by 299
  3. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
  4. 2022U-NO: U-shaped Neural Operatorscited by 219
  5. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
  6. 2024DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Trainingcited by 137
  7. 2023DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecastingcited by 105
  8. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  9. 2023Solving High-Dimensional PDEs with Latent Spectral Modelscited by 84
  10. 2023Operator Learning with Neural Fields: Tackling PDEs on General Geometriescited by 80
  11. 2023Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEscited by 66
  12. 2023Improved Operator Learning by Orthogonal Attentioncited by 58
38 more
  1. 2023Towards Stability of Autoregressive Neural Operatorscited by 54
  2. 2022Learning to correct spectral methods for simulating turbulent flowscited by 49
  3. 2024CViT: Continuous Vision Transformer for Operator Learningcited by 47
  4. 2022Mitigating spectral bias for the multiscale operator learningcited by 43
  5. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 42
  6. 2023Domain Agnostic Fourier Neural Operatorscited by 36
  7. 2023On the training and generalization of deep operator networkscited by 31
  8. 2024On the Benefits of Memory for Modeling Time-Dependent PDEscited by 30
  9. 2024DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systemscited by 29
  10. 2024UPS: Efficiently Building Foundation Models for PDE Solving via Cross-Modal Adaptationcited by 27
  11. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  12. 2024Universal Physics Transformers: A Framework For Efficiently Scaling Neural Operatorscited by 25
  13. 2024Text2PDE: Latent Diffusion Models for Accessible Physics Simulationcited by 25
  14. 2024HAMLET: Graph Transformer Neural Operator for Partial Differential Equationscited by 24
  15. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 24
  16. 2023A Neural PDE Solver with Temporal Stencil Modelingcited by 22
  17. 2023Harnessing the Power of Neural Operators with Automatically Encoded Conservation Lawscited by 22
  18. 2024BENO: Boundary-embedded Neural Operators for Elliptic PDEscited by 22
  19. 2022Guiding continuous operator learning through Physics-based boundary constraintscited by 21
  20. 2022Transform Once: Efficient Operator Learning in Frequency Domaincited by 20
  21. 2023Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domainscited by 20
  22. 2023Deep Equilibrium Based Neural Operators for Steady-State PDEscited by 19
  23. 2024Derivative-enhanced Deep Operator Networkcited by 19
  24. 2024PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systemscited by 18
  25. 2022NeurOLight: A Physics-Agnostic Neural Operator Enabling Parametric Photonic Device Simulationcited by 17
  26. 2024Mamba Neural Operator: Who Wins? Transformers vs. State-Space Models for PDEscited by 16
  27. 2023Deciphering and integrating invariants for neural operator learning with various physical mechanismscited by 15
  28. 2025Recurrent Neural Operators: Stable Long-Term PDE Predictioncited by 15
  29. 2023Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systemscited by 14
  30. 2023BubbleML: A Multi-Physics Dataset and Benchmarks for Machine Learningcited by 14
  31. 2024Knowledge-guided Machine Learning: Current Trends and Future Prospectscited by 13
  32. 2024SineNet: Learning Temporal Dynamics in Time-Dependent Partial Differential Equationscited by 13
  33. 2024P2C2Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamicscited by 13
  34. 2024DiffFNO: Diffusion Fourier Neural Operatorcited by 13
  35. 2024CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equationscited by 12
  36. 2024Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operatorcited by 12
  37. 2023Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representationcited by 11
  38. 2023Fourier Neural Operator Surrogate Model to Predict 3D Seismic Waves Propagationcited by 11

50 most cited of 237 All 237 (JSON)