Weakly-supervised Learning of Schrödinger Equation

We propose a machine learning method to solve Schrödinger 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.