Equilibrium Propagation for Periodic Dynamics

Equilibrium propagation provides an exact method to compute the gradient of a cost function from the response to external forcing. These algorithms, in conjunction with local updates to the system parameters, may enable mechanical and electronic systems to autonomously acquire complex functionality. We extend these methods to damped dynamical systems operating in the linear regime by introducing an effective action, whose extremum corresponds to the periodic steady state. The condition of applicability is that the response function is symmetric, which is fulfilled by damped Newtonian dynamics and RLC networks. We demonstrate the viability of our method in those systems and explore novel functionality such as classifying temporal sound signals. Our work opens the door to intelligent materials that process dynamical signals, enabling temporal computations, passive and active sensors, and materials that act as frequency-dependent filters.