Weakly-supervised Learning of Schrödinger Equation

We propose a machine learning method to solve Schr & ouml;dinger equations for Hamiltonians with perturbations. We focus on the cases where the unperturbed Hamiltonian can be solved analytically or numerically in some fast way. Using a perturbation potential function as input, our deep learning model predicts wave functions and energies efficiently. Training uses only the first-order perturbation energies on a training set to guide the convergence to the correction solutions. No label (or no exact solution) is used for the training, meaning that this is not a fully supervised method but rather a weakly supervised method since first-order perturbation energies are used to guide the learning. As an example, we calculated wave functions and energies of a harmonic oscillator with a perturbation, and the results were in good agreement with those obtained from exact diagonalization.