# Koopman neural operator as a mesh-free solver of non-linear partial differential equations

arXiv: https://arxiv.org/abs/2301.10022 · HTML: https://mlsanity.com/paper/2301.10022/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 102 references that are in the map.
- First posted on arXiv 24 Jan 2023.

## Builds on (most cited first, 30 of 33)

- [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)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [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)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [How Do Vision Transformers Work?](https://mlsanity.com/paper/2202.06709/) (2022)
- [Spectral Temporal Graph Neural Network for Multivariate Time-series Forecasting](https://mlsanity.com/paper/2103.07719/) (2021)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability](https://mlsanity.com/paper/1906.03663/) (2019)
- [Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition](https://mlsanity.com/paper/1409.5496/) (2014)
- [Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networks](https://mlsanity.com/paper/2203.08852/) (2022)

## Built on by (most cited first, 29 of 29)

- [Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors](https://mlsanity.com/paper/2305.18803/) (2023)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Improved Operator Learning by Orthogonal Attention](https://mlsanity.com/paper/2310.12487/) (2023)
- [CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamics](https://mlsanity.com/paper/2310.05963/) (2023)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning](https://mlsanity.com/paper/2408.13222/) (2024)
- [MetMamba: Regional Weather Forecasting with Spatial-Temporal Mamba Model](https://mlsanity.com/paper/2408.06400/) (2024)
- [RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations](https://mlsanity.com/paper/2509.01234/) (2025)
- [Capturing the Diffusive Behavior of the Multiscale Linear Transport Equations by Asymptotic-Preserving Convolutional DeepONets](https://mlsanity.com/paper/2306.15891/) (2023)
- [Space-Time Video Super-resolution with Neural Operator](https://mlsanity.com/paper/2404.06036/) (2024)
- [CoNO: Complex Neural Operator for Continous Dynamical Physical Systems](https://mlsanity.com/paper/2406.02597/) (2024)
- [PIC2O-Sim: A Physics-Inspired Causality-Aware Dynamic Convolutional Neural Operator for Ultra-Fast Photonic Device FDTD Simulation](https://mlsanity.com/paper/2406.17810/) (2024)
- [M2M: Learning controllable Multi of experts and multi-scale operators are the Partial Differential Equations need](https://mlsanity.com/paper/2410.11617/) (2024)
- [Learning Data-Efficient and Generalizable Neural Operators via Fundamental Physics Knowledge](https://mlsanity.com/paper/2602.15184/) (2026)
- [MeshONet: A Generalizable and Efficient Operator Learning Method for Structured Mesh Generation](https://mlsanity.com/paper/2501.11937/) (2025)
- [Invertible Koopman neural operator for data-driven modeling of partial differential equations](https://mlsanity.com/paper/2503.19717/) (2025)
- [Data-driven quantum Koopman method for simulating nonlinear dynamics](https://mlsanity.com/paper/2507.21890/) (2025)
- [Efficient High-Accuracy PDEs Solver with the Linear Attention Neural Operator](https://mlsanity.com/paper/2510.16816/) (2025)
- [Banach neural operator for Navier-Stokes equations](https://mlsanity.com/paper/2512.09070/) (2025)
- [Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators](https://mlsanity.com/paper/2605.10154/) (2026)
- [Deep Parallel Spectral Neural Operators for Solving Partial Differential Equations with Enhanced Low-Frequency Learning Capability](https://mlsanity.com/paper/2409.19976/) (2024)
- [Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices](https://mlsanity.com/paper/2410.02199/) (2024)
- [LE-PDE++: Mamba for accelerating PDEs Simulations](https://mlsanity.com/paper/2411.01897/) (2024)
- [A Neural Operator based on Dynamic Mode Decomposition](https://mlsanity.com/paper/2507.01117/) (2025)
- [SFO: Learning PDE Operators via Spectral Filtering](https://mlsanity.com/paper/2601.17090/) (2026)
- [PerFlow: Physics-Embedded Rectified Flow for Efficient Reconstruction and Uncertainty Quantification of Spatiotemporal Dynamics](https://mlsanity.com/paper/2605.03548/) (2026)
- [K-U-KAN: Koopman-Enhanced U-KAN for 3D Dental Reconstruction from a Single Panoramic X-ray Radiograph](https://mlsanity.com/paper/2605.25163/) (2026)
- [Let There Be Light: Reflection, Refraction and Scattering for Neural Operators](https://mlsanity.com/paper/2606.03262/) (2026)
- [Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis](https://mlsanity.com/paper/2609.04329/) (2026)

## Abstract (arXiv)

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural operators, a kind of neural-network-based PDE solver, these solvers become less accurate and explainable while learning long-term behaviors of non-linear PDE families. In this paper, we propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. With the same objective of learning an infinite-dimensional mapping between Banach spaces that serves as the solution operator of the target PDE family, our approach differs from existing models by formulating a non-linear dynamic system of equation solution. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, to act on the flow mapping of the dynamic system, we can equivalently learn the solution of a non-linear PDE family by solving simple linear prediction problems. We validate the KNO in mesh-independent, long-term, and5zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes equation and the Rayleigh-B'enard convection) and three real dynamic systems (e.g., global water vapor patterns and western boundary currents). In these experiments, the KNO exhibits notable advantages compared with previous state-of-the-art models, suggesting the potential of the KNO in supporting diverse science and engineering applications (e.g., PDE solving, turbulence modelling, and precipitation forecasting).