2023/02/02 by Elliott G. Holliday, Holliday, Elliott G., John F. Lindner +3 · 2 citations
Computer Science · Physics and Astronomy · #Computational Physics (physics.comp-ph) #FOS: Physical sciences #Model Reduction and Neural Networks #Neural Networks and Reservoir Computing #Quantum Physics (quant-ph) #Quantum, superfluid, helium dynamics
paper · pdf · doi:10.48550/arxiv.2302.01413
openalex publication_date 2023/02/02 · openalex created_date 2023/02/08 · openalex updated_date 2026/07/28
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 Schrödinger 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.