A Kernel Approach for PDE Discovery and Operator Learning
Machine Learning
2023-04-03 v2 Machine Learning
Abstract
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.
Cite
@article{arxiv.2210.08140,
title = {A Kernel Approach for PDE Discovery and Operator Learning},
author = {Da Long and Nicole Mrvaljevic and Shandian Zhe and Bamdad Hosseini},
journal= {arXiv preprint arXiv:2210.08140},
year = {2023}
}