2023/05/11 by Liam Harcombe, Harcombe, Liam, Quanling Deng +1 · 1 citation
Computer Science · Physics and Astronomy · #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Neural Networks and Applications #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2305.06802
openalex publication_date 2023/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.