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

A Kernel Approach for PDE Discovery and Operator Learning