  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.
