Deep learning neural network for approaching Schr\"odinger problems with arbitrary two-dimensional confinement

TL;DR

This paper introduces a neural network-based method to solve the two-dimensional Schrödinger equation for various confinement potentials, accurately predicting ground states without problem-specific tuning.

Contribution

It proposes a neural network architecture capable of generalizing to arbitrary 2D potentials for solving Schrödinger problems, demonstrating high accuracy across diverse cases.

Findings

High prediction accuracy for ground states in tested potentials

Effective generalization to unseen confinement potentials

Validation metrics confirm reliability of neural network estimates

Abstract

This article presents an approach to the two-dimensional Schr\"odinger equation based on automatic learning methods with neural networks. It is intended to determine the ground state of a particle confined in any two-dimensional potential, starting from the knowledge of the solutions to a large number of arbitrary sample problems. A network architecture with two hidden layers is proposed to predict the wave function and energy of the ground state. Several accuracy indicators are proposed for validating the estimates provided by the neural network. The testing of the trained network is done by applying it to a large set of confinement potentials different from those used in the learning process. Some particular cases with symmetrical potentials are solved as concrete examples, and a good network prediction accuracy is found.