Orthogonal greedy algorithm for linear operator learning with shallow neural network

Greedy algorithms, particularly the orthogonal greedy algorithm (OGA), have proven effective in training shallow neural networks for fitting functions and solving partial differential equations (PDEs). In this paper, we extend the application of OGA to the task of linear operator learning, which is equivalent to learning the kernel function through integral transforms. First, we develop a novel greedy algorithm for kernel estimation with respect to a new semi-inner product, enabling the approximation of the Green’s function for linear PDEs from data. Second, we introduce OGA-based point-wise kernel estimation to further improve the approximation rate, achieving orders of accuracy improvement across various tasks over baseline models. Additionally, we provide a theoretical analysis on the kernel estimation problem with the semi-inner product, deriving the optimal approximation rates for both algorithms. Our results demonstrate their efficacy and potential for future applications in PDE solving and operator learning.