Machine learning solutions for the two-dimensional quantum harmonic oscillator

Abstract

In this work, I have used Artificial Neural Networks to find the ground state of the 2D quantum harmonic oscillator. I have trained networks in two different ways: by using a mesh of points and by using Monte Carlo methods. I have used the analytical solution of the problem to benchmark the quality of the results of both methods, obtaining overlaps up to 0.99998 in the case of the mesh training and 0.9989 in the case of Monte Carlo. The relative errors in the energy are 0.03% and 1.1% respectively. I have shown the effects of the number of neurons and the learning rate on the overall performance of the network. Training with Monte Carlo shows faster convergence, while training on the mesh gets closer to the exact energy.