Physics-Informed Neural Networks for Discovering Localised Eigenstates in Disordered Media
This work addresses a longstanding problem in physics for modeling particle behavior in disordered systems, offering a machine-learning-based solution that is incremental in applying PINNs to this specific domain.
The paper tackles the challenge of discovering localised eigenstates in disordered media, which is difficult due to similar eigenenergies and high computational costs, by proposing a physics-informed neural network approach with a novel loss function that successfully identifies these states across various random potential distributions.
The Schrödinger 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 Schrödinger 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.