Neural network approaches for solving Schr\"odinger equation in arbitrary quantum wells

TL;DR

This paper introduces neural network models trained on classical solutions to approximate the Schrödinger equation in quantum wells with arbitrary potentials, demonstrating their accuracy and generalization capabilities.

Contribution

It proposes two neural network architectures trained on finite element solutions to efficiently solve the Schrödinger equation for arbitrary potentials.

Findings

Neural networks accurately estimate energies and wave functions.

Models generalize well to unseen potential data.

Comparison shows neural networks outperform classical methods in speed.

Abstract

In this work we approach the Schr\"odinger equation in quantum wells with arbitrary potentials, using the machine learning technique. Two neural networks with different architectures are proposed and trained using a set of potentials, energies, and wave functions previously generated with the classical finite element method. Three accuracy indicators have been proposed for testing the estimates given by the neural networks. The networks are trained by the gradient descent method and the training validation is done with respect to a large training data set. The two networks are then tested for two different potential data sets and the results are compared. Several cases with analytical potential have also been solved.