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.

翻译：具有随机势的薛定谔方程是理解无序系统中粒子行为的基本模型。无序介质的特征在于导致波函数局域化（也称为安德森局域化）的复杂势场。这些波函数的本征能量可能处于相近尺度，这给其发现带来了困难。由于求解薛定谔方程的高计算成本和高复杂度，这一问题长期存在挑战。近年来，机器学习工具已被用于应对这些挑战。本文基于机器学习的最新进展，提出了一种利用物理信息神经网络（PINNs）发现无序介质中局域本征态的新方法。我们聚焦于一维哈密顿量的谱近似，其势场根据伯努利分布、正态分布和均匀分布随机生成。我们在损失函数中引入了一个新特征，该特征利用这些区域中已知的物理现象，扫描整个定义域并成功发现这些本征态，无论其本征能量是否相近。我们通过多个算例展示了所提出方法的性能，并将其与等几何分析方法进行了比较。