mlsanityData to 7 Oct 2026

Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning

arXiv 2305.19913 ↗First posted 31 May 2023cs.LG eess.SPcited by 57

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Abstract (arXiv)

Recently, operator learning, or learning mappings between infinite-dimensional function spaces, has garnered significant attention, notably in relation to learning partial differential equations from data. Conceptually clear when outlined on paper, neural operators necessitate discretization in the transition to computer implementations. This step can compromise their integrity, often causing them to deviate from the underlying operators. This research offers a fresh take on neural operators with a framework Representation equivalent Neural Operators (ReNO) designed to address these issues. At its core is the concept of operator aliasing, which measures inconsistency between neural operators and their discrete representations. We explore this for widely-used operator learning techniques. Our findings detail how aliasing introduces errors when handling different discretizations and grids and loss of crucial continuous structures. More generally, this framework not only sheds light on existing challenges but, given its constructive and broad nature, also potentially offers tools for developing new neural operators.

Key facts. Cited by 57 papers in the map (data through 7 Oct 2026). Builds on 13 of its 25 references that are in the map. First posted on arXiv 31 May 2023.

Builds on

13 of 25 references.

References · 13

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  4. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  5. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  6. 2021Choose a Transformer: Fourier or Galerkincited by 333
  7. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  8. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
  9. 2022Learning Operators with Coupled Attentioncited by 88
  10. 2022NOMAD: Nonlinear Manifold Decoders for Operator Learningcited by 81
  11. 2022The Lie Derivative for Measuring Learned Equivariancecited by 51
  12. 2022Variable-Input Deep Operator Networkscited by 24
1 more
  1. 2022Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spacescited by 6

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 15; 2025: 22; 2026: 17.

57 papers.

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  1. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  2. 2024Poseidon: Efficient Foundation Models for PDEscited by 186
  3. 2024Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networkscited by 64
  4. 2023Plasma Surrogate Modelling using Fourier Neural Operatorscited by 29
  5. 2025Neural Chaos: A Spectral Stochastic Neural Operatorcited by 21
  6. 2024Basis-to-Basis Operator Learning Using Function Encoderscited by 17
  7. 2024Mixtures of Neural Operators Reduce Active Complexity in Operator Learningcited by 16
  8. 2024Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flowscited by 10
  9. 2024Multiple-Input Fourier Neural Operator (MIFNO) for source-dependent 3D elastodynamicscited by 8
  10. 2025Calibrated Physics-Informed Uncertainty Quantificationcited by 8
  11. 2025The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operatorscited by 8
  12. 2024FUSE: Fast Unified Simulation and Estimation for PDEscited by 6
38 more
  1. 2025When is a System Discoverable from Data? Discovery Requires Chaoscited by 6
  2. 2024Neural Operators Can Play Dynamic Stackelberg Gamescited by 5
  3. 2023Inverse Problems with Learned Forward Operatorscited by 4
  4. 2024Spectral Convolutional Conditional Neural Processescited by 4
  5. 2025Rectified Flows for Fast Multiscale Fluid Flow Modelingcited by 4
  6. 2024Structure-Preserving Operator Learningcited by 3
  7. 2025A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEscited by 3
  8. 2025Cauchy Random Features for Operator Learning in Sobolev Spacecited by 3
  9. 2025New universal operator approximation theorem for encoder-decoder architecturescited by 3
  10. 2026Imposing Boundary Conditions on Neural Operators via Learned Function Extensionscited by 3
  11. 2026Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithmcited by 3
  12. 2024Scientific Machine Learning Seismologycited by 2
  13. 2025Inverse Evolution Data Augmentation for Neural PDE Solverscited by 2
  14. 2025HyperNOs: Automated and Parallel Library for Neural Operators Researchcited by 2
  15. 2025Integrating Locality-Aware Attention with Transformers for General Geometry PDEscited by 2
  16. 2025RONOM: Reduced-Order Neural Operator Modelingcited by 2
  17. 2025ReBaNO: Reduced Basis Neural Operator Mitigating Generalization Gaps and Achieving Discretization Invariancecited by 2
  18. 2025Alias-Free ViT: Fractional Shift Invariance via Linear Attentioncited by 2
  19. 2025Convolutional-neural-operator-based transfer learning for solving PDEscited by 2
  20. 2025Can Data-Driven Dynamics Reveal Hidden Physics? There Is A Need for Interpretable Neural Operatorscited by 1
  21. 2025Breaking Scale Anchoring: Frequency Representation Learning for Accurate High-Resolution Inference from Low-Resolution Trainingcited by 1
  22. 2025Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Spacecited by 1
  23. 2026Adaptive Physics Transformer with Fused Global-Local Attention for Subsurface Energy Systemscited by 1
  24. 2026Learning Physical Operators using Neural Operatorscited by 1
  25. 2026NOIR: Neural Operator mapping for Implicit Representationscited by 1
  26. 2026GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domainscited by 1
  27. 2026WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domainscited by 1
  28. 2024A Resolution Independent Neural Operatorcited by 0
  29. 2024A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical resultscited by 0
  30. 2024Transfer Operator Learning with Fusion Framecited by 0
  31. 2024Adaptive Resolution Residual Networks -- Generalizing Across Resolutions Easily and Efficientlycited by 0
  32. 2025Mondrian: Transformer Operators via Domain Decompositioncited by 0
  33. 2025Interpretability and Generalization Bounds for Learning Spatial Physicscited by 0
  34. 2025Discretization-independent operator learning for partial differential equationscited by 0
  35. 2025Towards Multi-Fidelity Scaling Laws of Neural Surrogates in CFDcited by 0
  36. 2026The Error of Deep Operator Networks Is the Sum of Its Parts: Branch-Trunk and Mode Error Decompositionscited by 0
  37. 2026QuadNorm: Resolution-Robust Normalization for Neural Operatorscited by 0
  38. 2026Is Zero-Shot Super-Resolution Possible in Operator Learning?cited by 0

50 most cited of 57 All 57 (JSON)