Equilibrium Propagation for Learning in Lagrangian Dynamical Systems

TL;DR

This paper introduces a novel training method for Lagrangian dynamical systems using Equilibrium Propagation, enabling efficient learning without backpropagation through time, applicable to systems with various boundary conditions.

Contribution

It extends Equilibrium Propagation to dynamical systems governed by Lagrangian mechanics, allowing for parameter updates via trajectory nudging without explicit backpropagation.

Findings

Applicable to systems with periodic boundary conditions

Yields the semiclassical limit of Quantum Equilibrium Propagation

Addresses systems with dissipation

Abstract

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 how the variables conjugate to the parameters to be trained respond. 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.