Approximating Ground State Energies and Wave Functions of Physical Systems with Neural Networks

TL;DR

This paper introduces a deep learning method using neural networks to approximate ground state energies and wave functions of quantum systems by solving the Schrödinger equation variationally, demonstrating high accuracy on simple systems.

Contribution

It presents a novel end-to-end neural network approach for variationally solving the Schrödinger equation, applicable to complex physical systems beyond analytical solutions.

Findings

Achieves highly accurate ground state energy approximations

Successfully models wave functions for particle systems with perturbations

Demonstrates potential for complex quantum system solutions

Abstract

Quantum theory has been remarkably successful in providing an understanding of physical systems at foundational scales. Solving the Schr\"odinger equation provides full knowledge of all dynamical quantities of the physical system. However closed form solutions to this equation are only available for a few systems and approximation methods are typically used to find solutions. In this paper we address the problem of solving the time independent Schr\"odinger equation for the ground state solution of physical systems. We propose using end-to-end deep learning approach in a variational optimization scheme for approximating the ground state energies and wave functions of these systems. A neural network realizes a universal trial wave function and is trained in an unsupervised learning framework by optimizing the expectation value of the Hamiltonian of a physical system. The proposed approach is evaluated on physical systems consisting of a particle in a box with and without a perturbation. We demonstrate that our approach obtains approximations of ground state energies and wave functions that are highly accurate, which makes it a potentially plausible candidate for solving more complex physical systems for which analytical solutions are beyond reach.