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.

翻译：本文提出了一种基于核方法的三步框架，用于学习与求解偏微分方程（PDE）。给定一组由含噪声的PDE解及其在网格上的源项/边界项构成的训练对，首先采用核平滑技术对数据进行去噪，并逼近解的导数。随后，利用这些信息构建核回归模型，学习PDE的代数形式。最后，将学习得到的PDE应用于基于核的求解器中，从而在给定新源项/边界项时逼近PDE的解，由此构成一个算子学习框架。数值实验将该方法与现有最优算法进行了对比，验证了其具有竞争力的性能。