# A Kernel Approach for PDE Discovery and Operator Learning

arXiv: https://arxiv.org/abs/2210.08140 · HTML: https://mlsanity.com/paper/2210.08140/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 67 references that are in the map.
- First posted on arXiv 14 Oct 2022.

## Builds on (most cited first, 17 of 17)

- [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)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [A Probabilistic State Space Model for Joint Inference from Differential Equations and Data](https://mlsanity.com/paper/2103.10153/) (2021)
- [Gaussian Process Regression constrained by Boundary Value Problems](https://mlsanity.com/paper/2012.11857/) (2020)
- [Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations](https://mlsanity.com/paper/2208.12515/) (2022)
- [Probabilistic ODE Solutions in Millions of Dimensions](https://mlsanity.com/paper/2110.11812/) (2021)
- [Numerical methods for Mean field Games based on Gaussian Processes and Fourier Features](https://mlsanity.com/paper/2112.05414/) (2021)

## Built on by (most cited first, 7 of 7)

- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Multi-Resolution Active Learning of Fourier Neural Operators](https://mlsanity.com/paper/2309.16971/) (2023)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems](https://mlsanity.com/paper/2402.11722/) (2024)
- [Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulation](https://mlsanity.com/paper/2410.13794/) (2024)
- [Data-efficient Kernel Methods for Learning Hamiltonian Systems](https://mlsanity.com/paper/2509.17154/) (2025)
- [ROCK: A variational formulation for occupation kernel methods in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2503.13791/) (2025)

## Abstract (arXiv)

This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel smoothing is utilized to denoise the data and approximate derivatives of the solution. This information is then used in a kernel regression model to learn the algebraic form of the PDE. The learned PDE is then used within a kernel based solver to approximate the solution of the PDE with a new source/boundary term, thereby constituting an operator learning framework. Numerical experiments compare the method to state-of-the-art algorithms and demonstrate its competitive performance.