The Random Feature Method for Elliptic Eigenvalue Problems (Advances in Applied Mathematics and Mechanics, 2025)
Solving eigenvalue problems is an important subject in scientific computing. Classical numerical methods, such as finite element methods and spectral methods, have been developed with great success in applications. However, these methods struggle in the case of complex geometries. The recently developed random feature method has demonstrated its superiority in solving partial differential equations(PDEs), especially for problems with complex geometries. In this work, we developrandom feature methods for solving elliptic eigenvalue problems. Our contributionsinclude (1) using tailored separation-of-variables random feature functions to approximate eigenfunctions, (2) employing collocation points in the strong formulation orquadrature scheme (exact integration) in the weak formulation to handle the PDEs andboundary conditions, and (3) solving the generalized eigenvalue problem in which thenumber of conditions equals the number of unknowns. Through a series of examples from one dimension to three dimensions, we demonstrate the high accuracy androbustness of the proposed methods with respect to geometric complexity. We havemade our source code publicly available at https://github.com/lingyun-2024/The-code-of-RFM-EEP.git.