  The vast amount of design freedom in disordered systems expands the parameter
space for signal processing, allowing for unique signal flows that are
distinguished from those in regular systems. However, this large degree of
freedom has hindered the deterministic design of disordered systems for target
functionalities. Here, we employ a machine learning (ML) approach for
predicting and designing wave-matter interactions in disordered structures,
thereby identifying scale-free properties for waves. To abstract and map the
features of wave behaviours and disordered structures, we develop
disorder-to-localization and localization-to-disorder convolutional neural
networks (CNNs). Each CNN enables the instantaneous prediction of wave
localization in disordered structures and the instantaneous generation of
disordered structures from given localizations. We demonstrate that
CNN-generated disordered structures have scale-free properties with heavy tails
and hub atoms, which exhibit an increase of multiple orders of magnitude in
robustness to accidental defects, such as material or structural imperfection.
Our results verify the critical role of ML network structures in determining
ML-generated real-space structures, which can be used in the design of
defect-immune and efficiently tunable devices.
