# Neural Operators with Localized Integral and Differential Kernels

arXiv: https://arxiv.org/abs/2402.16845 · HTML: https://mlsanity.com/paper/2402.16845/

## Key facts

- Cited by 62 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 42 references that are in the map.
- First posted on arXiv 26 Feb 2024.

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

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [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)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [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)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Deep Parametric Continuous Convolutional Neural Networks](https://mlsanity.com/paper/2101.06742/) (2021)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Real-time high-resolution CO2 geological storage prediction using nested Fourier neural operators](https://mlsanity.com/paper/2210.17051/) (2022)
- [Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1411.6369/) (2014)
- [DISCO: accurate Discrete Scale Convolutions](https://mlsanity.com/paper/2106.02733/) (2021)
- [Scalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutions](https://mlsanity.com/paper/2209.13603/) (2022)
- [Scale Steerable Filters for Locally Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1906.03861/) (2019)
- [Scale equivariance in CNNs with vector fields](https://mlsanity.com/paper/1807.11783/) (2018)
- [Truly Scale-Equivariant Deep Nets with Fourier Layers](https://mlsanity.com/paper/2311.02922/) (2023)
- [Broadband Ground Motion Synthesis via Generative Adversarial Neural Operators: Development and Validation](https://mlsanity.com/paper/2309.03447/) (2023)

## Built on by (most cited first, 30 of 62)

- [FourCastNet 3: A geometric approach to probabilistic machine-learning weather forecasting at scale](https://mlsanity.com/paper/2507.12144/) (2025)
- [CViT: Continuous Vision Transformer for Operator Learning](https://mlsanity.com/paper/2405.13998/) (2024)
- [A Library for Learning Neural Operators](https://mlsanity.com/paper/2412.10354/) (2024)
- [DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic prediction](https://mlsanity.com/paper/2504.19496/) (2025)
- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Mesh-Informed Neural Operator : A Transformer Generative Approach](https://mlsanity.com/paper/2506.16656/) (2025)
- [Enhancing Fourier Neural Operators with Local Spatial Features](https://mlsanity.com/paper/2503.17797/) (2025)
- [Enabling Local Neural Operators to perform Equation-Free System-Level Analysis](https://mlsanity.com/paper/2505.02308/) (2025)
- [Attention on the Sphere](https://mlsanity.com/paper/2505.11157/) (2025)
- [Universal Functional Regression with Neural Operator Flows](https://mlsanity.com/paper/2404.02986/) (2024)
- [Turb-L1: Achieving Long-term Turbulence Tracing By Tackling Spectral Bias](https://mlsanity.com/paper/2505.19038/) (2025)
- [Exploring the design space of deep-learning-based weather forecasting systems](https://mlsanity.com/paper/2410.07472/) (2024)
- [A Unified Model for Compressed Sensing MRI Across Undersampling Patterns](https://mlsanity.com/paper/2410.16290/) (2024)
- [KANO: Kolmogorov-Arnold Neural Operator](https://mlsanity.com/paper/2509.16825/) (2025)
- [Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators](https://mlsanity.com/paper/2508.00643/) (2025)
- [SAOT: An Enhanced Locality-Aware Spectral Transformer for Solving PDEs](https://mlsanity.com/paper/2511.18777/) (2025)
- [Diffeomorphic Latent Neural Operators for Data-Efficient Learning of Solutions to Partial Differential Equations](https://mlsanity.com/paper/2411.18014/) (2024)
- [Continuous Representation Methods, Theories, and Applications: An Overview and Perspectives](https://mlsanity.com/paper/2505.15222/) (2025)
- [Differential-Integral Neural Operator for Long-Term Turbulence Forecasting](https://mlsanity.com/paper/2509.21196/) (2025)
- [Liquid Fourier Latent Dynamics Networks for fast GPU-based numerical simulations in computational cardiology](https://mlsanity.com/paper/2408.09818/) (2024)
- [Physics-Aware Neural Operators for Direct Inversion in 3D Photoacoustic Tomography](https://mlsanity.com/paper/2509.09894/) (2025)
- [Generalized Spherical Neural Operators: Green's Function Formulation](https://mlsanity.com/paper/2512.10723/) (2025)
- [Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reduction](https://mlsanity.com/paper/2407.03925/) (2024)
- [Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems](https://mlsanity.com/paper/2409.03302/) (2024)
- [Learning Greens Operators through Hierarchical Neural Networks Inspired by the Fast Multipole Method](https://mlsanity.com/paper/2509.20591/) (2025)
- [DRIFT-Net: A Spectral--Coupled Neural Operator for PDEs Learning](https://mlsanity.com/paper/2509.24868/) (2025)
- [PDE Solvers Should Be Local: Fast, Stable Rollouts with Learned Local Stencils](https://mlsanity.com/paper/2509.26186/) (2025)
- [Function-Space Decoupled Diffusion for Forward and Inverse Modeling in Carbon Capture and Storage](https://mlsanity.com/paper/2602.12274/) (2026)
- [HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs](https://mlsanity.com/paper/2605.00820/) (2026)
- [Accelerating PDE-Constrained Optimization by the Derivative of Neural Operators](https://mlsanity.com/paper/2506.13120/) (2025)

## Abstract (arXiv)

Neural operators learn mappings between function spaces, which is practical for learning solution operators of PDEs and other scientific modeling applications. Among them, the Fourier neural operator (FNO) is a popular architecture that performs global convolutions in the Fourier space. However, such global operations are often prone to over-smoothing and may fail to capture local details. In contrast, convolutional neural networks (CNN) can capture local features but are limited to training and inference at a single resolution. In this work, we present a principled approach to operator learning that can capture local features under two frameworks by learning differential operators and integral operators with locally supported kernels. Specifically, inspired by stencil methods, we prove that we obtain differential operators under an appropriate scaling of the kernel values of CNNs. To obtain local integral operators, we utilize suitable basis representations for the kernels based on discrete-continuous convolutions. Both these approaches preserve the properties of operator learning and, hence, the ability to predict at any resolution. Adding our layers to FNOs significantly improves their performance, reducing the relative L2-error by 34-72% in our experiments, which include a turbulent 2D Navier-Stokes and the spherical shallow water equations.