# CoNO: Complex Neural Operator for Continous Dynamical Physical Systems

arXiv: https://arxiv.org/abs/2406.02597 · HTML: https://mlsanity.com/paper/2406.02597/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 75 references that are in the map.
- First posted on arXiv 1 Jun 2024.

## 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)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [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)
- [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)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Fast Sampling of Diffusion Models via Operator Learning](https://mlsanity.com/paper/2211.13449/) (2022)
- [Full-Capacity Unitary Recurrent Neural Networks](https://mlsanity.com/paper/1611.00035/) (2016)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Koopman Operator Dynamical Models: Learning, Analysis and Control](https://mlsanity.com/paper/2102.02522/) (2021)
- [Associative Long Short-Term Memory](https://mlsanity.com/paper/1602.03032/) (2016)
- [Complex Transformer: A Framework for Modeling Complex-Valued Sequence](https://mlsanity.com/paper/1910.10202/) (2019)
- [Koopman neural operator as a mesh-free solver of non-linear partial differential equations](https://mlsanity.com/paper/2301.10022/) (2023)
- [Neuronal Synchrony in Complex-Valued Deep Networks](https://mlsanity.com/paper/1312.6115/) (2013)
- [The universal approximation theorem for complex-valued neural networks](https://mlsanity.com/paper/2012.03351/) (2020)
- [Convolutional Complex Knowledge Graph Embeddings](https://mlsanity.com/paper/2008.03130/) (2020)

## Built on by (most cited first, 3 of 3)

- [Time Series Analysis in Frequency Domain: A Survey of Open Challenges, Opportunities and Benchmarks](https://mlsanity.com/paper/2504.07099/) (2025)
- [Latent Mamba Operator for Partial Differential Equations](https://mlsanity.com/paper/2505.19105/) (2025)
- [TF-SNO: Time-Frequency Gated Spectral Neural Operators for Learning Non-Stationary Partial Differential Equations](https://mlsanity.com/paper/2606.21189/) (2026)

## Abstract (arXiv)

Neural operators extend data-driven models to map between infinite-dimensional functional spaces. While these operators perform effectively in either the time or frequency domain, their performance may be limited when applied to non-stationary spatial or temporal signals whose frequency characteristics change with time. Here, we introduce Complex Neural Operator (CoNO) that parameterizes the integral kernel using Fractional Fourier Transform (FrFT), better representing non-stationary signals in a complex-valued domain. Theoretically, we prove the universal approximation capability of CoNO. We perform an extensive empirical evaluation of CoNO on seven challenging partial differential equations (PDEs), including regular grids, structured meshes, and point clouds. Empirically, CoNO consistently attains state-of-the-art performance, showcasing an average relative gain of 10.9%. Further, CoNO exhibits superior performance, outperforming all other models in additional tasks such as zero-shot super-resolution and robustness to noise. CoNO also exhibits the ability to learn from small amounts of data -- giving the same performance as the next best model with just 60% of the training data. Altogether, CoNO presents a robust and superior model for modeling continuous dynamical systems, providing a fillip to scientific machine learning.