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

arXiv: https://arxiv.org/abs/2305.19913 · HTML: https://mlsanity.com/paper/2305.19913/

## Key facts

- Cited by 57 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 25 references that are in the map.
- First posted on arXiv 31 May 2023.

## Builds on (most cited first, 13 of 13)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [The Lie Derivative for Measuring Learned Equivariance](https://mlsanity.com/paper/2210.02984/) (2022)
- [Variable-Input Deep Operator Networks](https://mlsanity.com/paper/2205.11404/) (2022)
- [Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces](https://mlsanity.com/paper/2205.14627/) (2022)

## Built on by (most cited first, 30 of 57)

- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks](https://mlsanity.com/paper/2406.11045/) (2024)
- [Plasma Surrogate Modelling using Fourier Neural Operators](https://mlsanity.com/paper/2311.05967/) (2023)
- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (2024)
- [Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows](https://mlsanity.com/paper/2405.17211/) (2024)
- [Multiple-Input Fourier Neural Operator (MIFNO) for source-dependent 3D elastodynamics](https://mlsanity.com/paper/2404.10115/) (2024)
- [FUSE: Fast Unified Simulation and Estimation for PDEs](https://mlsanity.com/paper/2405.14558/) (2024)
- [Calibrated Physics-Informed Uncertainty Quantification](https://mlsanity.com/paper/2502.04406/) (2025)
- [The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operators](https://mlsanity.com/paper/2510.06646/) (2025)
- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [Neural Operators Can Play Dynamic Stackelberg Games](https://mlsanity.com/paper/2411.09644/) (2024)
- [Cauchy Random Features for Operator Learning in Sobolev Space](https://mlsanity.com/paper/2503.00300/) (2025)
- [Inverse Problems with Learned Forward Operators](https://mlsanity.com/paper/2311.12528/) (2023)
- [Spectral Convolutional Conditional Neural Processes](https://mlsanity.com/paper/2404.13182/) (2024)
- [New universal operator approximation theorem for encoder-decoder architectures](https://mlsanity.com/paper/2503.24092/) (2025)
- [Rectified Flows for Fast Multiscale Fluid Flow Modeling](https://mlsanity.com/paper/2506.03111/) (2025)
- [Structure-Preserving Operator Learning](https://mlsanity.com/paper/2410.01065/) (2024)
- [A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEs](https://mlsanity.com/paper/2502.08683/) (2025)
- [Imposing Boundary Conditions on Neural Operators via Learned Function Extensions](https://mlsanity.com/paper/2602.04923/) (2026)
- [Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithm](https://mlsanity.com/paper/2602.05052/) (2026)
- [Scientific Machine Learning Seismology](https://mlsanity.com/paper/2409.18397/) (2024)
- [Inverse Evolution Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2501.14604/) (2025)
- [HyperNOs: Automated and Parallel Library for Neural Operators Research](https://mlsanity.com/paper/2503.18087/) (2025)
- [Integrating Locality-Aware Attention with Transformers for General Geometry PDEs](https://mlsanity.com/paper/2504.13480/) (2025)
- [RONOM: Reduced-Order Neural Operator Modeling](https://mlsanity.com/paper/2507.12814/) (2025)
- [ReBaNO: Reduced Basis Neural Operator Mitigating Generalization Gaps and Achieving Discretization Invariance](https://mlsanity.com/paper/2509.09611/) (2025)
- [Alias-Free ViT: Fractional Shift Invariance via Linear Attention](https://mlsanity.com/paper/2510.22673/) (2025)

## 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.