Solving two-dimensional quantum eigenvalue problems using physics-informed machine learning

TL;DR

This paper demonstrates how physics-informed neural networks can efficiently solve two-dimensional quantum eigenvalue problems for various geometries, extending methods from 1D to more complex 2D systems.

Contribution

It introduces a generalized unsupervised learning algorithm using neural networks to find eigenvalues and eigenfunctions in 2D quantum systems with diverse boundary shapes.

Findings

Successfully solves 2D quantum eigenvalue problems for multiple geometries.

Generalizes neural network approach to Helmholtz and wave equations.

Provides a versatile method applicable to quantum chaos and electromagnetic cavity modes.

Abstract

A particle confined to an impassable box is a paradigmatic and exactly solvable one-dimensional quantum system modeled by an infinite square well potential. Here we explore some of its infinitely many generalizations to two dimensions, including particles confined to rectangle, elliptic, triangle, and cardioid-shaped boxes, using physics-informed neural networks. In particular, we generalize an unsupervised learning algorithm to find the particles' eigenvalues and eigenfunctions. During training, the neural network adjusts its weights and biases, one of which is the energy eigenvalue, so its output approximately solves the Schr\"odinger equation with normalized and mutually orthogonal eigenfunctions. The same procedure solves the Helmholtz equation for the harmonics and vibration modes of waves on drumheads or transverse magnetic modes of electromagnetic cavities. Related applications include dynamical billiards, quantum chaos, and Laplacian spectra.