Deep Learning Solution of the Eigenvalue Problem for Differential Operators
Authors: Ido Ben-Shaul, Leah Bar, Nir Sochen
Keywords: Eigenvalue problem, Unsupervised learning, Laplacian operator
Abstract: Solving the eigenvalue problem for differential operators is a common problem in many scientific fields. Classical numerical methods rely on intricate domain discretization, and yield non-analytic or non-smooth approximations. We introduce a novel Neural Network (NN)-based solver for the eigenvalue problem of differential self-adjoint operators where the eigenpairs are learned in an unsupervised end-to-end fashion. We propose three different training procedures, for solving increasingly challenging tasks towards the general eigenvalue problem. 

The proposed solver is able to find the M smallest eigenpairs for a general differential operator. We demonstrate the method on the Laplacian operator which is of particular interest in image processing, computer vision, shape analysis among many other applications. 
Unlike other numerical methods such as finite differences, the partial derivatives of the network approximation of the eigenfunction can be analytically calculated to any order. Therefore, the proposed framework enables the solution of higher order operators and on free shape domain or even on a manifold. Non-linear operators can be investigated by this approach as well.