TL;DR
Convolutional Neural Operator (CNO) is an operator-learning architecture that uses convolutional/multi-resolution components to map between function spaces. It aims to retain the efficiency of CNNs while behaving like an operator model that can generalize across grids/resolutions.
Problem
Fourier-based operators (like FNO) excel on periodic/regular grids but can be sensitive to resolution changes and boundary geometry. CNO targets operator learning with convolutional building blocks and multi-scale resampling that better accommodate varied discretizations.
Benefits vs others
- Convolutional inductive bias can be strong for local PDE structure; multi-scale mixing helps capture long-range effects.
- Often easier to deploy on image-like PDE grids and potentially more memory-friendly than attention-based models.
- Can be used as a competitive operator-learning baseline alongside FNO/DeepONet.
Interesting detail
- CNO is useful when Fourier assumptions are questionable (non-periodic boundaries, irregular sampling) or when you want CNN tooling.
- It provides a clean operator-learning baseline for PDEBench-style datasets.
Core method (math)
Template for Operator learning. Paper-specific equations are added when manually curated.
\[u = \mathcal{G}_\theta(a)\quad\text{(learn an operator mapping input field }a\text{ to output field }u\text{)}\]
\[\text{(Convolutional operator block)}\;\; v_{l+1} = \sigma\!\left(W_lv_l + (K_l * v_l)\right)\]
\[\text{(Multi-resolution mixing)}\;\; v \xrightarrow{\downarrow} v^{(s)} \xrightarrow{\text{conv}} \xrightarrow{\uparrow} v\]
Main theoretical contribution
- CNO approximates integral operators via learned convolutional kernels and multi-scale aggregation.
- Resolution robustness is pursued through explicit resampling pathways (down/up operators).
Main contribution
- Defines an operator-learning architecture using only convolutions (no global Fourier mixing).
- Benchmarks across a broad PDE suite (Poisson, Wave, Navier–Stokes, etc.).
- Provides strong baseline performance with simple operations and good scaling.
Main results (headline)
(Optional) Add main_results for a quick headline summary.
Experiments
PDE problems
- Poisson
- Wave equation
- Navier–Stokes
- Allen–Cahn
- Transport
- Compressible Euler
- Darcy flow
Tasks
- Forward operator learning
- Resolution generalization
Experiment setting (high level)
- Evaluated on Representative PDE Benchmarks (RPB) and related datasets.
- Reports accuracy and efficiency across PDE types.
Comparable baselines
- FNO
- U-NO
- WNO
- U-Net
Main results
Key results
| PDE suite | Metric | Reported takeaway |
|---|---|---|
| Multiple PDEs | L2 / rollout error | Competitive accuracy across a wide PDE set with convolution-only design. |
Citation (BibTeX)
@article{cno2023,
title={Convolutional Neural Operator for Robust Operator Learning},
author={Kovachki, Nikola and others},
journal={arXiv preprint arXiv:2302.01178},
year={2023}
}