# On the locality of local neural operator in learning fluid dynamics

arXiv: https://arxiv.org/abs/2312.09820 · HTML: https://mlsanity.com/paper/2312.09820/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 47 references that are in the map.
- First posted on arXiv 15 Dec 2023.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [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)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Understanding the Effective Receptive Field in Deep Convolutional Neural Networks](https://mlsanity.com/paper/1701.04128/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Deep transfer operator learning for partial differential equations under conditional shift](https://mlsanity.com/paper/2204.09810/) (2022)
- [Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads](https://mlsanity.com/paper/2306.08218/) (2023)
- [Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads](https://mlsanity.com/paper/2306.03645/) (2023)
- [Stochastic projection based approach for gradient free physics informed learning](https://mlsanity.com/paper/2209.13724/) (2022)
- [Bayesian Physics Informed Neural Networks for Data Assimilation and Spatio-Temporal Modelling of Wildfires](https://mlsanity.com/paper/2212.00970/) (2022)
- [Local neural operator for solving transient partial differential equations on varied domains](https://mlsanity.com/paper/2203.08145/) (2022)

## Built on by (most cited first, 6 of 6)

- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modeling](https://mlsanity.com/paper/2409.08477/) (2024)
- [Generalized Spherical Neural Operators: Green's Function Formulation](https://mlsanity.com/paper/2512.10723/) (2025)
- [GALDS: A Graph-Autoencoder-based Latent Dynamics Surrogate model to predict neurite material transport](https://mlsanity.com/paper/2507.10871/) (2025)
- [PDE Solvers Should Be Local: Fast, Stable Rollouts with Learned Local Stencils](https://mlsanity.com/paper/2509.26186/) (2025)
- [Virtual domain extension for imposing boundary conditions in flow simulation using pre-trained local neural operator](https://mlsanity.com/paper/2504.09807/) (2025)

## Abstract (arXiv)

This paper launches a thorough discussion on the locality of local neural operator (LNO), which is the core that enables LNO great flexibility on varied computational domains in solving transient partial differential equations (PDEs). We investigate the locality of LNO by looking into its receptive field and receptive range, carrying a main concern about how the locality acts in LNO training and applications. In a large group of LNO training experiments for learning fluid dynamics, it is found that an initial receptive range compatible with the learning task is crucial for LNO to perform well. On the one hand, an over-small receptive range is fatal and usually leads LNO to numerical oscillation; on the other hand, an over-large receptive range hinders LNO from achieving the best accuracy. We deem rules found in this paper general when applying LNO to learn and solve transient PDEs in diverse fields. Practical examples of applying the pre-trained LNOs in flow prediction are presented to confirm the findings further. Overall, with the architecture properly designed with a compatible receptive range, the pre-trained LNO shows commendable accuracy and efficiency in solving practical cases.