Low-Fidelity FDM Spectral Guidance for Neural Eigenvalue Solvers

Operator eigenvalue problems appear throughout science. Classical methods usually discretize the operator into a matrix and then solve the resulting matrix eigenvalue problem. This works well in low dimensions, but fine grids quickly become expensive in both memory and computation as the dimension grows. Neural network based solvers avoid storing these large grids, but recent state of the art neural methods can require hundreds of thousands of training steps and may struggle to find the desired eigenvalues. We show that the two approaches can help each other. A coarse finite difference method (FDM) calculation acts as a cheap numerical model of the operator spectrum. We use the approximate eigenvalues as fixed shifts during the training of the neural solver, as they only need to locate the relevant part of the spectrum. We also introduce Stabilized Inverse Power Method Neural Network (SIPMNN), a more stable training procedure for higher-dimensional problems. Across five test problems at $d=10$, the combined approach is more accurate overall than the tested fully neural alternatives while using eight to ten times fewer iterations.