Neural Network Acceleration of Iterative Methods for Nonlinear Schr\"odinger Eigenvalue Problems

Abstract

We present a novel approach to accelerate iterative methods to solve nonlinear Schr\"odinger eigenvalue problems using neural networks. Nonlinear eigenvector problems are fundamental in quantum mechanics and other fields, yet conventional solvers often suffer from slow convergence in extreme parameter regimes, as exemplified by the rotating Bose- Einstein condensate (BEC) problem. Our method uses a neural network to predict and refine solution trajectories, leveraging knowledge from previous simulations to improve convergence speed and accuracy. Numerical experiments demonstrate significant speed-up over classical solvers, highlighting both the strengths and limitations of the approach.