Kernel-based learning methods for stochastic partial differential equations

This article delves into the study of kernel-based learning methods for stochastic partial differential equations. The theory of generalized data and kernel-based probability measures is introduced to construct kernel-based learning estimators, kernel-based learning functions, and discrete kernel-based learning solutions for addressing stochastic differentials, elliptic stochastic partial differential equations, and parabolic stochastic partial differential equations, respectively. The convergence theorems of kernel-based learning algorithms are demonstrated by combining meshfree approximation and kriging interpolation. Moreover, the numerical examples show the efficiency and robustness of kernel-based learning algorithms using various positive definite kernels.