# Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learning

arXiv: https://arxiv.org/abs/2407.00809 · HTML: https://mlsanity.com/paper/2407.00809/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 86 references that are in the map.
- First posted on arXiv 30 Jun 2024.

## Builds on (most cited first, 30 of 34)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [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)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Gaussian Process Kernels for Pattern Discovery and Extrapolation](https://mlsanity.com/paper/1302.4245/) (2013)
- [Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs](https://mlsanity.com/paper/2310.00120/) (2023)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [Non-Stationary Spectral Kernels](https://mlsanity.com/paper/1705.08736/) (2017)
- [Non-Stationary Gaussian Process Regression with Hamiltonian Monte Carlo](https://mlsanity.com/paper/1508.04319/) (2015)
- [Spectral Mixture Kernels for Multi-Output Gaussian Processes](https://mlsanity.com/paper/1709.01298/) (2017)
- [RiemannONets: Interpretable Neural Operators for Riemann Problems](https://mlsanity.com/paper/2401.08886/) (2024)
- [Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizations](https://mlsanity.com/paper/2205.09332/) (2022)
- [BCR-Net: a neural network based on the nonstandard wavelet form](https://mlsanity.com/paper/1810.08754/) (2018)
- [Meta-learning Pseudo-differential Operators with Deep Neural Networks](https://mlsanity.com/paper/1906.06782/) (2019)
- [Ensemble and Mixture-of-Experts DeepONets For Operator Learning](https://mlsanity.com/paper/2405.11907/) (2024)
- [Neural Non-Stationary Spectral Kernel](https://mlsanity.com/paper/1811.10978/) (2018)

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

- [Operator Learning with Gaussian Processes](https://mlsanity.com/paper/2409.04538/) (2024)
- [Optimal Convergence Rates for Neural Operators](https://mlsanity.com/paper/2412.17518/) (2024)
- [Polynomial Chaos Expansion for Operator Learning](https://mlsanity.com/paper/2508.20886/) (2025)
- [Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulation](https://mlsanity.com/paper/2410.13794/) (2024)
- [Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flows](https://mlsanity.com/paper/2602.15472/) (2026)
- [Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learning](https://mlsanity.com/paper/2606.17513/) (2026)
- [FlowDA: Accurate, Low-Latency Weather Data Assimilation via Flow Matching](https://mlsanity.com/paper/2602.06800/) (2026)
- [Enabling Real-Time Training of a Wildfire-to-Smoke Map with Multilinear Operators](https://mlsanity.com/paper/2605.04164/) (2026)
- [Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data](https://mlsanity.com/paper/2607.02203/) (2026)
- [Quadrature-Aware Complex-Linear Neural Operator for Boundary-to-Field Prediction in Resonant Acoustics](https://mlsanity.com/paper/2607.04407/) (2026)
- [The Frame Kernel Method for Multiscale Operator Learning](https://mlsanity.com/paper/2608.25084/) (2026)

## Abstract (arXiv)

This paper introduces the Kernel Neural Operator (KNO), a provably convergent operator-learning architecture that utilizes compositions of deep kernel-based integral operators for function-space approximation of operators (maps from functions to functions). The KNO decouples the choice of kernel from the numerical integration scheme (quadrature), thereby naturally allowing for operator learning with explicitly-chosen trainable kernels on irregular geometries. On irregular domains, this allows the KNO to utilize domain-specific quadrature rules. To help ameliorate the curse of dimensionality, we also leverage an efficient dimension-wise factorization algorithm on regular domains. More importantly, the ability to explicitly specify kernels also allows the use of highly expressive, non-stationary, neural anisotropic kernels whose parameters are computed by training neural networks. We present universal approximation theorems showing that both the continuous and fully discretized KNO are universal approximators on operator learning problems. Numerical results demonstrate that on existing benchmarks the training and test accuracy of KNOs is closely comparable to or higher than that of popular neural operators while typically using an order of magnitude fewer trainable parameters, with the more expressive kernels proving important to attaining high accuracy. KNOs thus facilitate low-memory, geometrically-flexible, deep operator learning, while retaining the implementation simplicity and transparency of traditional kernel methods from both scientific computing and machine learning.