A Kernel Approach for PDE Discovery and Operator Learning
This work addresses the challenge of PDE discovery and operator learning for computational science and engineering applications, representing an incremental advancement by combining kernel methods in a novel framework.
The authors tackled the problem of learning and solving partial differential equations (PDEs) from noisy data by developing a three-step kernel-based framework that denoises data, learns PDE forms, and approximates solutions, showing competitive performance in numerical experiments compared to state-of-the-art methods.
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.