Insights into adaptive learning of general dynamical behaviors

Physical learning is a growing field studying physical systems that can adapt themselves to learn desired properties without the use of a central processor. In particular, by observing labeled data, these adaptable systems can adjust their effective interactions to attain global physical features, such as atypical auxetic properties in mechanical metamaterials, allosteric responses in networks, or even classification and regression tasks. The systems rely on physically-achievable local learning rules that update a system's adaptable degrees of freedom with access only to locally available information. To date, this program has primarily focused on systems operating in equilibrium or steady-state, such that any externally imposed input and corresponding desired physical response are both static. Here, we address the problem of physical learning with dynamical inputs and dynamical desired functionalities, both for equilibrium systems under external, time-dependent driving or, more generally, for non-equilibrium dynamical systems. Using ideas from previously successful learning schemes, such as Coupled Learning and Equilibrium Propagation, we study a general local learning rule for dynamical behaviors. We demonstrate our results for important model systems such as coupled Kuramoto oscillators and dynamically driven networks.