  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.
