Computing gradients of a cost function is central to design-based optimization and machine learning algorithms. Equilibrium propagation provides an exact method to compute gradients in hardware by exploiting the inherent physical laws. The locality of these algorithms, in conjunction with local updates, enables mechanical and electronic systems that autonomously learn a function. We extend these methods to damped dynamical systems operating in the linear regime, such as mechanical structures obeying damped Newtonian dynamics and RLC circuits. By introducing an effective action whose extremum corresponds to the underlying dynamics, we derive local learning rules. This approach applies both to problems with periodic boundary conditions and to those with resting initial conditions. We demonstrate the viability of our method in mechanical and electronic 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.
Berneman et al. (Wed,) studied this question.