mlsanityData to 7 Oct 2026

U-NO: U-shaped Neural Operators

arXiv 2204.11127 ↗First posted 23 Apr 2022cs.LGcited by 219

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Abstract (arXiv)

Neural operators generalize classical neural networks to maps between infinite-dimensional spaces, e.g., function spaces. Prior works on neural operators proposed a series of novel methods to learn such maps and demonstrated unprecedented success in learning solution operators of partial differential equations. Due to their close proximity to fully connected architectures, these models mainly suffer from high memory usage and are generally limited to shallow deep learning models. In this paper, we propose U-shaped Neural Operator (U-NO), a U-shaped memory enhanced architecture that allows for deeper neural operators. U-NOs exploit the problem structures in function predictions and demonstrate fast training, data efficiency, and robustness with respect to hyperparameters choices. We study the performance of U-NO on PDE benchmarks, namely, Darcy's flow law and the Navier-Stokes equations. We show that U-NO results in an average of 26% and 44% prediction improvement on Darcy's flow and turbulent Navier-Stokes equations, respectively, over the state of the art. On Navier-Stokes 3D spatiotemporal operator learning task, we show U-NO provides 37% improvement over the state of art methods.

Key facts. Cited by 219 papers in the map (data through 7 Oct 2026). Builds on 25 of its 33 references that are in the map. First posted on arXiv 23 Apr 2022.

Builds on

25 of 33 references.

References · 25

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13 more
  1. 2017Exploring Generalization in Deep Learningcited by 595
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  6. 2021Factorized Fourier Neural Operatorscited by 237
  7. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  8. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  9. 2019Symplectic Recurrent Neural Networkscited by 148
  10. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  11. 2021A Farewell to the Bias-Variance Tradeoff? An Overview of the Theory of Overparameterized Machine Learningcited by 54
  12. 2021Seismic wave propagation and inversion with Neural Operatorscited by 51
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 9; 2023: 31; 2024: 43; 2025: 58; 2026: 78.

219 papers.

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  1. 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 293
  2. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
  3. 2023Spherical Fourier Neural Operators: Learning Stable Dynamics on the Spherecited by 225
  4. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  5. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
  6. 2023Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behaviorcited by 128
  7. 2022Clifford Neural Layers for PDE Modelingcited by 91
  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. 2024Operator Learning: Algorithms and Analysiscited by 84
  11. 2023Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEscited by 66
  12. 2022Generative Adversarial Neural Operatorscited by 51
38 more
  1. 2024CViT: Continuous Vision Transformer for Operator Learningcited by 47
  2. 2022Mitigating spectral bias for the multiscale operator learningcited by 43
  3. 2023MgNO: Efficient Parameterization of Linear Operators via Multigridcited by 40
  4. 2023Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generationcited by 37
  5. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 33
  6. 2024On the Benefits of Memory for Modeling Time-Dependent PDEscited by 30
  7. 2023Koopman neural operator as a mesh-free solver of non-linear partial differential equationscited by 29
  8. 2024RiemannONets: Interpretable Neural Operators for Riemann Problemscited by 28
  9. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  10. 2023Rethinking materials simulations: Blending direct numerical simulations with neural operatorscited by 25
  11. 2022Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEscited by 24
  12. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 24
  13. 2023Variational Autoencoding Neural Operatorscited by 22
  14. 2023Harnessing the Power of Neural Operators with Automatically Encoded Conservation Lawscited by 22
  15. 2024Active Learning for Neural PDE Solverscited by 22
  16. 2023∞-Diff: Infinite Resolution Diffusion with Subsampled Mollified Statescited by 21
  17. 2024PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systemscited by 18
  18. 2025Probabilistic neural operators for functional uncertainty quantificationcited by 17
  19. 2023Real-time Inference and Extrapolation via a Diffusion-inspired Temporal Transformer Operator (DiTTO)cited by 16
  20. 2023Deciphering and integrating invariants for neural operator learning with various physical mechanismscited by 15
  21. 2024Accelerating PDE Data Generation via Differential Operator Action in Solution Spacecited by 15
  22. 2023BubbleML: A Multi-Physics Dataset and Benchmarks for Machine Learningcited by 14
  23. 2025TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolationcited by 14
  24. 2024A hybrid iterative method based on MIONet for PDEs: Theory and numerical examplescited by 13
  25. 2024SineNet: Learning Temporal Dynamics in Time-Dependent Partial Differential Equationscited by 13
  26. 2023Broadband Ground Motion Synthesis via Generative Adversarial Neural Operators: Development and Validationcited by 12
  27. 2024Accelerating Data Generation for Neural Operators via Krylov Subspace Recyclingcited by 12
  28. 2025MORPH: PDE Foundation Models with Arbitrary Data Modalitycited by 12
  29. 2023Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representationcited by 11
  30. 2023Fourier Neural Operator Surrogate Model to Predict 3D Seismic Waves Propagationcited by 11
  31. 2024Generating synthetic data for neural operatorscited by 11
  32. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 11
  33. 2025Mesh-Informed Neural Operator : A Transformer Generative Approachcited by 11
  34. 2023Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspectivecited by 10
  35. 2023Equivariant Neural Operator Learning with Graphon Convolutioncited by 10
  36. 2024Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flowscited by 10
  37. 2024Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Spacecited by 10
  38. 2025CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operatorscited by 10

50 most cited of 219 All 219 (JSON)