# Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations

arXiv: https://arxiv.org/abs/2303.02304 · HTML: https://mlsanity.com/paper/2303.02304/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 62 references that are in the map.
- First posted on arXiv 4 Mar 2023.

## Builds on (most cited first, 20 of 20)

- [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)
- [Scheduled Sampling for Sequence Prediction with Recurrent Neural Networks](https://mlsanity.com/paper/1506.03099/) (2015)
- [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)
- [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)
- [Spectral Temporal Graph Neural Network for Multivariate Time-series Forecasting](https://mlsanity.com/paper/2103.07719/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [A physics-informed operator regression framework for extracting data-driven continuum models](https://mlsanity.com/paper/2009.11992/) (2020)
- [Spectral Temporal Graph Neural Network for Trajectory Prediction](https://mlsanity.com/paper/2106.02930/) (2021)
- [Counterfactual Neural Temporal Point Process for Estimating Causal Influence of Misinformation on Social Media](https://mlsanity.com/paper/2210.07518/) (2022)
- [Proximal methods for stationary Mean Field Games with local couplings](https://mlsanity.com/paper/1608.07701/) (2016)
- [On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings](https://mlsanity.com/paper/1802.07902/) (2018)
- [Splitting methods for a class of non-potential mean field games](https://mlsanity.com/paper/2007.00099/) (2020)
- [DSLOB: A Synthetic Limit Order Book Dataset for Benchmarking Forecasting Algorithms under Distributional Shift](https://mlsanity.com/paper/2211.11513/) (2022)

## Built on by (most cited first, 11 of 11)

- [TimeDiT: General-purpose Diffusion Transformers for Time Series Foundation Model](https://mlsanity.com/paper/2409.02322/) (2024)
- [BENO: Boundary-embedded Neural Operators for Elliptic PDEs](https://mlsanity.com/paper/2401.09323/) (2024)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Multi-scale Conditional Generative Modeling for Microscopic Image Restoration](https://mlsanity.com/paper/2407.05259/) (2024)
- [COMPOL: A Unified Neural Operator Framework for Scalable Multi-Physics Simulations](https://mlsanity.com/paper/2501.17296/) (2025)
- [Fast 3D Surrogate Modeling for Data Center Thermal Management](https://mlsanity.com/paper/2511.11722/) (2025)
- [Adaptive Mamba Neural Operators](https://mlsanity.com/paper/2607.18043/) (2026)
- [Extending Fourier Neural Operators for Modeling Parameterized and Coupled PDEs](https://mlsanity.com/paper/2607.23466/) (2026)
- [Neuro-Inspired Hierarchical Multimodal Learning](https://mlsanity.com/paper/2309.15877/) (2023)
- [Theory Guided and Interpretable Neural Operator Design for Partial Differential Equation Learning](https://mlsanity.com/paper/2609.33715/) (2026)
- [A Generative Model of Complex Networks Using Graphons and Neural Inverse Operators](https://mlsanity.com/paper/2610.02439/) (2026)

## Abstract (arXiv)

Coupled partial differential equations (PDEs) are key tasks in modeling the complex dynamics of many physical processes. Recently, neural operators have shown the ability to solve PDEs by learning the integral kernel directly in Fourier/Wavelet space, so the difficulty for solving the coupled PDEs depends on dealing with the coupled mappings between the functions. Towards this end, we propose a textitcoupled multiwavelets neural operator (CMWNO) learning scheme by decoupling the coupled integral kernels during the multiwavelet decomposition and reconstruction procedures in the Wavelet space. The proposed model achieves significantly higher accuracy compared to previous learning-based solvers in solving the coupled PDEs including Gray-Scott (GS) equations and the non-local mean field game (MFG) problem. According to our experimental results, the proposed model exhibits a 2× ~ 4× improvement relative L2 error compared to the best results from the state-of-the-art models.