Physics-Informed Neural Networks for Discovering Localised Eigenstates in Disordered Media

TL;DR

This paper introduces a physics-informed neural network method to efficiently discover localized eigenstates in disordered media, overcoming computational challenges and accurately identifying eigenstates with similar energies.

Contribution

The paper presents a novel PINN-based approach with a new loss function feature for discovering localized eigenstates in disordered systems, improving over traditional methods.

Findings

Successfully discovers localized eigenstates in disordered media

Performs well across different random potential distributions

Outperforms isogeometric analysis in accuracy and efficiency

Abstract

The Schr\"{o}dinger equation with random potentials is a fundamental model for understanding the behaviour of particles in disordered systems. Disordered media are characterised by complex potentials that lead to the localisation of wavefunctions, also called Anderson localisation. These wavefunctions may have similar scales of eigenenergies which poses difficulty in their discovery. It has been a longstanding challenge due to the high computational cost and complexity of solving the Schr\"{o}dinger equation. Recently, machine-learning tools have been adopted to tackle these challenges. In this paper, based upon recent advances in machine learning, we present a novel approach for discovering localised eigenstates in disordered media using physics-informed neural networks (PINNs). We focus on the spectral approximation of Hamiltonians in one dimension with potentials that are randomly generated according to the Bernoulli, normal, and uniform distributions. We introduce a novel feature to the loss function that exploits known physical phenomena occurring in these regions to scan across the domain and successfully discover these eigenstates, regardless of the similarity of their eigenenergies. We present various examples to demonstrate the performance of the proposed approach and compare it with isogeometric analysis.