mlsanityData to 7 Oct 2026

Convolutional Neural Operators for robust and accurate learning of PDEs

arXiv 2302.01178 ↗First posted 2 Feb 2023cs.LGcited by 207

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Abstract (arXiv)

Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of PDEs. Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs. The resulting architecture, termed as convolutional neural operators (CNOs), is designed specifically to preserve its underlying continuous nature, even when implemented in a discretized form on a computer. We prove a universality theorem to show that CNOs can approximate operators arising in PDEs to desired accuracy. CNOs are tested on a novel suite of benchmarks, encompassing a diverse set of PDEs with possibly multi-scale solutions and are observed to significantly outperform baselines, paving the way for an alternative framework for robust and accurate operator learning. Our code is publicly available at https://github.com/bogdanraonic3/ConvolutionalNeuralOperator

Key facts. Cited by 207 papers in the map (data through 7 Oct 2026). Builds on 38 of its 67 references that are in the map. First posted on arXiv 2 Feb 2023.

Builds on

38 of 67 references.

References · 38

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  10. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  11. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  12. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
26 more
  1. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  2. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  3. 2021ResNet strikes back: An improved training procedure in timmcited by 365
  4. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  5. 2017Stable Architectures for Deep Neural Networkscited by 340
  6. 2021Choose a Transformer: Fourier or Galerkincited by 333
  7. 2017PDE-Net: Learning PDEs from Datacited by 304
  8. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  9. 2021Factorized Fourier Neural Operatorscited by 237
  10. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  11. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
  12. 2021Deep Parametric Continuous Convolutional Neural Networkscited by 161
  13. 2022Learning Operators with Coupled Attentioncited by 88
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  21. 2018Equivalence of approximation by convolutional neural networks and fully-connected networkscited by 38
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  23. 2022Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuitiescited by 29
  24. 2020Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networkscited by 26
  25. 2022Variable-Input Deep Operator Networkscited by 24
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 49; 2025: 74; 2026: 81.

207 papers.

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  1. 2024Poseidon: Efficient Foundation Models for PDEscited by 186
  2. 2024The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learningcited by 134
  3. 2024Operator Learning: Algorithms and Analysiscited by 84
  4. 2023Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learningcited by 57
  5. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  6. 2024Wavelet Diffusion Neural Operatorcited by 28
  7. 2025Generative Latent Neural PDE Solver using Flow Matchingcited by 28
  8. 2023An operator preconditioning perspective on training in physics-informed machine learningcited by 27
  9. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  10. 2025Walrus: A Cross-Domain Foundation Model for Continuum Dynamicscited by 26
  11. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 24
  12. 2024Discretization Error of Fourier Neural Operatorscited by 20
38 more
  1. 2023AI Foundation Models for Weather and Climate: Applications, Design, and Implementationcited by 19
  2. 2025Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulationcited by 19
  3. 2024Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problemscited by 18
  4. 2024FlowBench: A Large Scale Benchmark for Flow Simulation over Complex Geometriescited by 18
  5. 2024Mixtures of Neural Operators Reduce Active Complexity in Operator Learningcited by 16
  6. 2025TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolationcited by 14
  7. 2024Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solverscited by 13
  8. 2025AgenticSciML: Collaborative Multi-Agent Systems for Emergent Discovery in Scientific Machine Learningcited by 13
  9. 2026RealPDEBench: A Benchmark for Complex Physical Systems with Real-World Datacited by 13
  10. 2024Generating synthetic data for neural operatorscited by 11
  11. 2024Hierarchical Neural Operator Transformer with Learnable Frequency-aware Loss Prior for Arbitrary-scale Super-resolutioncited by 11
  12. 2024Model Balancing Helps Low-data Training and Fine-tuningcited by 11
  13. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 11
  14. 2024Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flowscited by 10
  15. 2024Neural Green's Operators for Parametric Partial Differential Equationscited by 10
  16. 2024Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbonecited by 10
  17. 2024Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methodscited by 10
  18. 2024Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Spacecited by 10
  19. 2025Turb-L1: Achieving Long-term Turbulence Tracing By Tackling Spectral Biascited by 9
  20. 2024OmniArch: Building Foundation Model For Scientific Computingcited by 8
  21. 2025Enhancing Fourier Neural Operators with Local Spatial Featurescited by 8
  22. 2025The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operatorscited by 8
  23. 2026Test-time Generalization for Physics through Neural Operator Splittingcited by 8
  24. 2024A Unified Model for Compressed Sensing MRI Across Undersampling Patternscited by 7
  25. 2025TANTE: Time-Adaptive Operator Learning via Neural Taylor Expansioncited by 7
  26. 2024FUSE: Fast Unified Simulation and Estimation for PDEscited by 6
  27. 2024An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learningcited by 6
  28. 2024A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problemscited by 6
  29. 2024Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous mediacited by 5
  30. 2024PACE: Pacing Operator Learning to Accurate Optical Field Simulation for Complicated Photonic Devicescited by 5
  31. 2024Neural Operators Can Play Dynamic Stackelberg Gamescited by 5
  32. 2025A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometriescited by 5
  33. 2025Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogatescited by 5
  34. 2026Diagnosing Failure Modes of Neural Operators Across Diverse PDE Familiescited by 5
  35. 2026Universal Diffusion-Based Probabilistic Downscalingcited by 5
  36. 2024Learning on manifolds without manifold learningcited by 4
  37. 2024Spectral Convolutional Conditional Neural Processescited by 4
  38. 2024Separable Operator Networkscited by 4

50 most cited of 207 All 207 (JSON)