# KernelOnet: An Interpretable Neural Operator Based on Kernel Functions

arXiv: https://arxiv.org/abs/2609.35938 · HTML: https://mlsanity.com/paper/2609.35938/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 55 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 14 of 14)

- [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)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (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)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithm](https://mlsanity.com/paper/2602.05052/) (2026)

## Abstract (arXiv)

This paper proposes an interpretable neural operator framework, the Kernel Operator Network (KernelOnet), which incorporates kernel functions explicitly into the neural operator architecture, so that the operator structure matches the kernel-expansion form used in boundary-type kernel-expansion methods. Unlike traditional neural operators such as DeepONet, which learn basis functions implicitly through deep networks, KernelOnet replaces the trunk network with explicit kernels and offers three complementary kernels: a data-driven learnable kernel, in which a neural network parameterizes a radial basis function learned from data, and which for constant-coefficient linear problems can be regarded as a non-singular fundamental solution; a physics-informed kernel, which embeds physical information such as analytic fundamental solutions into the network structure, so that the expansion satisfies the governing equation automatically and can be trained without supervision on boundary conditions alone, with no interior solution data; and a hybrid kernel, which splits the solution, according to the linear principal part of the governing equation, into a homogeneous part spanned by analytic fundamental solutions and a source part carried by low-rank learned correction kernels, thereby balancing physical priors against data fitting on nonlinear problems lacking an analytic fundamental solution. On three benchmarks and one engineering problem in a shallow-water waveguide, KernelOnet attains high accuracy; where comparable with DeepONet, it is more accurate with fewer learnable parameters. Its unsupervised configuration needs no interior solution labels, and its per-query inference cost is far below that of per-instance solvers, offering an effective route to acoustic propagation in unbounded exterior domains that general-purpose neural operators struggle to handle.