# Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs

arXiv: https://arxiv.org/abs/2204.06255 · HTML: https://mlsanity.com/paper/2204.06255/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 19 references that are in the map.
- First posted on arXiv 13 Apr 2022.

## Builds on (most cited first, 7 of 7)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (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)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [A physics-informed operator regression framework for extracting data-driven continuum models](https://mlsanity.com/paper/2009.11992/) (2020)
- [Feature Engineering with Regularity Structures](https://mlsanity.com/paper/2108.05879/) (2021)

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

- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [Transformer Meets Boundary Value Inverse Problems](https://mlsanity.com/paper/2209.14977/) (2022)
- [A Brief Survey on the Approximation Theory for Sequence Modelling](https://mlsanity.com/paper/2302.13752/) (2023)
- [Better Neural PDE Solvers Through Data-Free Mesh Movers](https://mlsanity.com/paper/2312.05583/) (2023)
- [Stochastic Latent Transformer: Efficient Modelling of Stochastically Forced Zonal Jets](https://mlsanity.com/paper/2310.16741/) (2023)
- [Infinite-dimensional reservoir computing](https://mlsanity.com/paper/2304.00490/) (2023)
- [Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equations](https://mlsanity.com/paper/2601.01021/) (2026)
- [Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experiments](https://mlsanity.com/paper/2507.17887/) (2025)
- [SPDEBench: An Extensive Benchmark for Learning Stochastic PDEs](https://mlsanity.com/paper/2505.18511/) (2025)
- [Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equations](https://mlsanity.com/paper/2603.08219/) (2026)
- [Learning Manifold and Ito Dynamics with Branched Neural Rough Differential Equations](https://mlsanity.com/paper/2606.05272/) (2026)

## Abstract (arXiv)

Stochastic partial differential equations (SPDEs) are significant tools for modeling dynamics in many areas including atmospheric sciences and physics. Neural Operators, generations of neural networks with capability of learning maps between infinite-dimensional spaces, are strong tools for solving parametric PDEs. However, they lack the ability to modeling SPDEs which usually have poor regularity due to the driving noise. As the theory of regularity structure has achieved great successes in analyzing SPDEs and provides the concept model feature vectors that well-approximate SPDEs' solutions, we propose the Neural Operator with Regularity Structure (NORS) which incorporates the feature vectors for modeling dynamics driven by SPDEs. We conduct experiments on various of SPDEs including the dynamic Phi41 model and the 2d stochastic Navier-Stokes equation, and the results demonstrate that the NORS is resolution-invariant, efficient, and achieves one order of magnitude lower error with a modest amount of data.