Orthogonal greedy algorithm for linear operator learning with shallow neural network

TL;DR

This paper extends the orthogonal greedy algorithm to linear operator learning, enabling efficient kernel estimation for PDEs and improving approximation accuracy with theoretical guarantees.

Contribution

It introduces a novel greedy algorithm for kernel estimation in a new semi-inner product and applies OGA for point-wise kernel estimation, enhancing accuracy and providing theoretical analysis.

Findings

Achieved orders of accuracy improvement in kernel approximation tasks.

Established theoretical optimal approximation rates for the proposed algorithms.

Demonstrated effectiveness in approximating Green's functions for linear PDEs.

Abstract

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 tasks of linear operator learning, which is equivalent to learning the kernel function through integral transforms. Firstly, a novel greedy algorithm is developed for kernel estimation rate in a new semi-inner product, which can be utilized to approximate the Green's function of linear PDEs from data. Secondly, we introduce the OGA for point-wise kernel estimation to further improve the approximation rate, achieving orders of accuracy improvement across various tasks and baseline models. In addition, we provide a theoretical analysis on the kernel estimation problem and the optimal approximation rates for both algorithms, establishing their efficacy and potential for future applications in PDEs and operator learning tasks.