Machine learning imprints scale-free networks on disordered materials

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 propose a machine learning (ML) approach for predicting and designing wave-matter interactions in disordered structures, thereby revealing scale-free networks for waves. To abstract and map the features of wave behaviours and disordered structures, we develop matter-to-wave and wave-to-matter 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 operate as scale-free "wave" networks with hub atoms, which exhibit an increase of multiple orders of magnitude in robustness to accidental attacks. Our results verify the critical role of ML network structures in determining ML-generated real-space structures, which can be used in the network-inspired design of wave systems.