Weakly-supervised learning on Schrodinger equation

TL;DR

This paper introduces a weakly-supervised deep learning approach to solve Schrödinger equations for perturbed Hamiltonians, leveraging perturbation theory without requiring exact solutions for training.

Contribution

The novel method trains neural networks using perturbation information, enabling efficient calculation of wave functions and energies for arbitrary perturbations without labeled data.

Findings

Accurately predicted wave functions and energies for a perturbed harmonic oscillator.

Good agreement with exact diagonalization results.

Applicable to Hamiltonians with arbitrary perturbations.

Abstract

We propose a machine learning method to solve Schrodinger equations for a Hamiltonian that consists of an unperturbed Hamiltonian and a perturbation. We focus on the cases where the unperturbed Hamiltonian can be solved analytically or solved numerically with some fast way. Given a potential function as input, our deep learning model predicts wave functions and energies using a weakly-supervised method. Information of first-order perturbation calculation for randomly chosen perturbations is used to train the model. In other words, no label (or exact solution) is necessary for the training, which is why the method is called weakly-supervised, not supervised. The trained model can be applied to calculation of wave functions and energies of Hamiltonian containing arbitrary perturbation. As an example, we calculated wave functions and energies of a harmonic oscillator with a perturbation and results were in good agreement with those obtained from exact diagonalization.