Equilibrium propagation for learning in Lagrangian dynamical systems

We propose a method for training dynamical systems governed by Lagrangian mechanics using equilibrium propagation. Our approach extends equilibrium propagation-initially developed for energy-based models-to dynamical trajectories by leveraging the principle of action extremization. Training is achieved by gently nudging trajectories toward desired targets and measuring the response of the variables conjuguate to the trainable parameters. This method is particularly suited to systems with periodic boundary conditions or fixed initial and final states, enabling efficient parameter updates without requiring explicit backpropagation through time. In the case of periodic boundary conditions, this approach yields the semiclassical limit of quantum equilibrium propagation. Applications to systems with dissipation are also discussed.