mlsanityData to 14 Oct 2026

Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere

arXiv 2306.03838 ↗First posted 6 Jun 2023cs.LG cs.NA math.NAcited by 233

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Abstract (arXiv)

Fourier Neural Operators (FNOs) have proven to be an efficient and effective method for resolution-independent operator learning in a broad variety of application areas across scientific machine learning. A key reason for their success is their ability to accurately model long-range dependencies in spatio-temporal data by learning global convolutions in a computationally efficient manner. To this end, FNOs rely on the discrete Fourier transform (DFT), however, DFTs cause visual and spectral artifacts as well as pronounced dissipation when learning operators in spherical coordinates since they incorrectly assume a flat geometry. To overcome this limitation, we generalize FNOs on the sphere, introducing Spherical FNOs (SFNOs) for learning operators on spherical geometries. We apply SFNOs to forecasting atmospheric dynamics, and demonstrate stable auto-regressive rollouts for a year of simulated time (1,460 steps), while retaining physically plausible dynamics. The SFNO has important implications for machine learning-based simulation of climate dynamics that could eventually help accelerate our response to climate change.

Key facts. Cited by 233 papers in the map (data through 14 Oct 2026). Builds on 20 of its 53 references that are in the map. First posted on arXiv 6 Jun 2023.

Builds on

20 of 53 references.

References · 20

  1. 2020An Image is Worth 16x16 Words: Transformers for Image Recognition at Scalecited by 30,186
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  3. 2016Gaussian Error Linear Units (GELUs)cited by 3,502
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
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  8. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,167
  9. 2016Group Equivariant Convolutional Networkscited by 1,165
  10. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 776
  11. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  12. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 297
8 more
  1. 2020WeatherBench: A benchmark dataset for data-driven weather forecastingcited by 275
  2. 2021Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformerscited by 248
  3. 2022U-NO: U-shaped Neural Operatorscited by 230
  4. 2022Pangu-Weather: A 3D High-Resolution Model for Fast and Accurate Global Weather Forecastcited by 166
  5. 2020Improving data-driven global weather prediction using deep convolutional neural networks on a cubed spherecited by 96
  6. 2021Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction modelscited by 79
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  8. 2022Scalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutionscited by 24

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 20; 2024: 59; 2025: 74; 2026: 80.

233 papers.

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  1. 2023WeatherBench 2: A benchmark for the next generation of data-driven global weather modelscited by 236
  2. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 221
  3. 2024Poseidon: Efficient Foundation Models for PDEscited by 200
  4. 2023Residual Corrective Diffusion Modeling for Km-scale Atmospheric Downscalingcited by 110
  5. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 108
  6. 2024Operator Learning: Algorithms and Analysiscited by 85
  7. 2024Neural Operators with Localized Integral and Differential Kernelscited by 62
  8. 2025FourCastNet 3: A geometric approach to probabilistic machine-learning weather forecasting at scalecited by 62
  9. 2024Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEscited by 59
  10. 2023ACE: A fast, skillful learned global atmospheric model for climate predictioncited by 56
  11. 2024ACE2: Accurately learning subseasonal to decadal atmospheric variability and forced responsescited by 54
  12. 2024A Library for Learning Neural Operatorscited by 38
38 more
  1. 2025OneForecast: A Universal Framework for Global and Regional Weather Forecastingcited by 34
  2. 2024On the Foundations of Earth and Climate Foundation Modelscited by 33
  3. 2024Kilometer-Scale Convection Allowing Model Emulation using Generative Diffusion Modelingcited by 32
  4. 2024DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systemscited by 30
  5. 2024LUCIE: A Lightweight Uncoupled ClImate Emulator with long-term stability and physical consistency for O(1000)-member ensemblescited by 30
  6. 2023Advancing Parsimonious Deep Learning Weather Prediction using the HEALPix Meshcited by 28
  7. 2025Fixing the Double Penalty in Data-Driven Weather Forecasting Through a Modified Spherical Harmonic Loss Functioncited by 28
  8. 2023ClimateSet: A Large-Scale Climate Model Dataset for Machine Learningcited by 25
  9. 2024A Practical Probabilistic Benchmark for AI Weather Modelscited by 25
  10. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 25
  11. 2023Graph-based Neural Weather Prediction for Limited Area Modelingcited by 22
  12. 2023AI Foundation Models for Weather and Climate: Applications, Design, and Implementationcited by 21
  13. 2024Coupled Ocean-Atmosphere Dynamics in a Machine Learning Earth System Modelcited by 21
  14. 2024Probabilistic Emulation of a Global Climate Model with Spherical DYffusioncited by 21
  15. 2024Huge Ensembles Part I: Design of Ensemble Weather Forecasts using Spherical Fourier Neural Operatorscited by 18
  16. 2025Probabilistic neural operators for functional uncertainty quantificationcited by 18
  17. 2025Geometric Operator Learning with Optimal Transportcited by 16
  18. 2024Analyzing and Exploring Training Recipes for Large-Scale Transformer-Based Weather Predictioncited by 15
  19. 2026Demystifying Data-Driven Probabilistic Medium-Range Weather Forecastingcited by 15
  20. 2025CirT: Global Subseasonal-to-Seasonal Forecasting with Geometry-inspired Transformercited by 14
  21. 2025Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equationscited by 14
  22. 2023Do AI models produce better weather forecasts than physics-based models? A quantitative evaluation case study of Storm Ciar'ancited by 12
  23. 2024Linearization Turns Neural Operators into Function-Valued Gaussian Processescited by 12
  24. 2024MANTRA: The Manifold Triangulations Assemblagecited by 12
  25. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 12
  26. 2026U-Cast: A Surprisingly Simple and Efficient Frontier Probabilistic AI Weather Forecastercited by 12
  27. 2024Neural Green's Operators for Parametric Partial Differential Equationscited by 11
  28. 2024LD-EnSF: Synergizing Latent Dynamics with Ensemble Score Filters for Fast Data Assimilation with Sparse Observationscited by 11
  29. 2025Diffusion-LAM: Probabilistic Limited Area Weather Forecasting with Diffusioncited by 11
  30. 2025FlashBias: Fast Computation of Attention with Biascited by 11
  31. 2023Forecasting Tropical Cyclones with Cascaded Diffusion Modelscited by 10
  32. 2024GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applicationscited by 10
  33. 2024Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Spacecited by 10
  34. 2025Denoising Hamiltonian Network for Physical Reasoningcited by 10
  35. 2025Enhancing Fourier Neural Operators with Local Spatial Featurescited by 10
  36. 2025Attention on the Spherecited by 10
  37. 2023Guaranteed Approximation Bounds for Mixed-Precision Neural Operatorscited by 9
  38. 2023FrePolad: Frequency-Rectified Point Latent Diffusion for Point Cloud Generationcited by 9

50 most cited of 233 All 233 (JSON)