# Convolutional Neural Operators for robust and accurate learning of PDEs

arXiv: https://arxiv.org/abs/2302.01178 · HTML: https://mlsanity.com/paper/2302.01178/

## Key facts

- Cited by 214 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 67 references that are in the map.
- First posted on arXiv 2 Feb 2023.

## Builds on (most cited first, 30 of 38)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [ResNet strikes back: An improved training procedure in timm](https://mlsanity.com/paper/2110.00476/) (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)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Deep Parametric Continuous Convolutional Neural Networks](https://mlsanity.com/paper/2101.06742/) (2021)
- [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)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [Pad'e Activation Units: End-to-end Learning of Flexible Activation Functions in Deep Networks](https://mlsanity.com/paper/1907.06732/) (2019)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)

## Built on by (most cited first, 30 of 214)

- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning](https://mlsanity.com/paper/2305.19913/) (2023)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Walrus: A Cross-Domain Foundation Model for Continuum Dynamics](https://mlsanity.com/paper/2511.15684/) (2025)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [Generative Latent Neural PDE Solver using Flow Matching](https://mlsanity.com/paper/2503.22600/) (2025)
- [An operator preconditioning perspective on training in physics-informed machine learning](https://mlsanity.com/paper/2310.05801/) (2023)
- [Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspective](https://mlsanity.com/paper/2404.07200/) (2024)
- [Discretization Error of Fourier Neural Operators](https://mlsanity.com/paper/2405.02221/) (2024)
- [AI Foundation Models for Weather and Climate: Applications, Design, and Implementation](https://mlsanity.com/paper/2309.10808/) (2023)
- [Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems](https://mlsanity.com/paper/2402.11722/) (2024)
- [FlowBench: A Large Scale Benchmark for Flow Simulation over Complex Geometries](https://mlsanity.com/paper/2409.18032/) (2024)
- [Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulation](https://mlsanity.com/paper/2507.02608/) (2025)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (2024)
- [RealPDEBench: A Benchmark for Complex Physical Systems with Real-World Data](https://mlsanity.com/paper/2601.01829/) (2026)
- [Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solvers](https://mlsanity.com/paper/2405.15557/) (2024)
- [TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation](https://mlsanity.com/paper/2505.17341/) (2025)
- [AgenticSciML: Collaborative Multi-Agent Systems for Emergent Discovery in Scientific Machine Learning](https://mlsanity.com/paper/2511.07262/) (2025)
- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Generating synthetic data for neural operators](https://mlsanity.com/paper/2401.02398/) (2024)
- [Hierarchical Neural Operator Transformer with Learnable Frequency-aware Loss Prior for Arbitrary-scale Super-resolution](https://mlsanity.com/paper/2405.12202/) (2024)
- [Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows](https://mlsanity.com/paper/2405.17211/) (2024)
- [Neural Green's Operators for Parametric Partial Differential Equations](https://mlsanity.com/paper/2406.01857/) (2024)
- [Model Balancing Helps Low-data Training and Fine-tuning](https://mlsanity.com/paper/2410.12178/) (2024)
- [Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbone](https://mlsanity.com/paper/2406.17119/) (2024)
- [Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methods](https://mlsanity.com/paper/2410.06820/) (2024)
- [Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space](https://mlsanity.com/paper/2410.11382/) (2024)

## 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