Sustaining the chaotic dynamics of the Kuramoto model by adaptable reservoir computer

A scenario frequently encountered in real-world complex systems is the temporary failure of a few components. For spatially extended chaotic systems whose functionality hinges on the collective dynamics of the interacting components, a viable approach to dealing with the failures is replacing the malfunctioning components with their backups, so that the collective dynamics of the systems can be precisely maintained for a duration long enough to resolve the problem. Here, taking the chaotic dynamics of the paradigmatic Kuramoto model as an example and considering the scenario of oscillator failures, we propose substituting the failed oscillators with digital twins trained by the data measured from the system evolution. Specifically, leveraging the technique of an adaptable reservoir computer (RC) in machine learning, we demonstrate that a single, small-size RC can substitute any oscillator in the Kuramoto model such that the time evolution of the synchronization order parameter of the repaired system is identical to that of the original system for a certain time period. The performance of adaptable RC is evaluated in various contexts, and it is found that the sustaining period is influenced by multiple factors, including the size of the training dataset, the overall coupling strength of the system, and the number of substituted oscillators. Additionally, though the synchronization order parameter diverges from the ground truth in the long-term running, the functional networks of the oscillators are still faithfully sustained by the machine substitutions.