Cluster synchronization of reservoir computers in complex network

Abstract Synchronization pervades natural and engineered systems across multiple scales. For systems lacking explicit governing equations, machine learning provides a data-driven approach to reconstruct and analyze their synchronizable dynamics. While previous studies have primarily realized global synchronization among coupled machine-learning oscillators, the emergence of cluster synchronization—a more intricate collective behavior—has remained unverified in such purely data-driven systems. Here, we demonstrate and validate cluster synchronization in neural networks composed of reservoir computers (RCs) that operate without predefined physical models. By mixing the readout signals according to the network connectivity, RCs within the same cluster autonomously converge to identical trajectories,whereas those in different clusters remain desynchronized. Two characteristic collective modes-isolated desynchronization and intertwined synchronization-are identified and analyzed.These results reveal that symmetry-induced cluster synchronization can spontaneously emerge in machine-learned dynamical networks, establishing RC systems as a powerful platform for exploring collective behaviors and constructing digital twins of complex physical systems beyond explicit modeling.