Equilibrium Propagation for Learning in Lagrangian Dynamical Systems

2025-05-12

AI Summary

Backpropagation through time (BPTT) suffers from high computational overhead and struggles with continuous-time trajectories—especially in dynamical systems subject to periodic boundary conditions or fixed endpoint constraints. Method: This paper introduces a Lagrangian action-based equilibrium propagation framework, leveraging the principle of extremal action. Gradients are estimated directly by perturbing system trajectories and analyzing the steady-state response of conjugate variables to parameter perturbations—bypassing explicit backward-in-time computation. Contributions/Results: First, equilibrium propagation is extended to continuous-time dynamical trajectories. Second, its semiclassical limit under periodic boundaries is derived, unifying quantum and classical variational learning perspectives. Third, the framework accommodates both fixed initial/final states and dissipative dynamics. Experiments demonstrate substantial improvements over BPTT in gradient accuracy, training stability, and computational efficiency—establishing a new paradigm for physics-informed temporal modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• Extends Equilibrium Propagation to Lagrangian dynamical systems
• Trains systems without explicit backpropagation through time
• Suitable for periodic or fixed boundary condition systems

Innovation

• Extends Equilibrium Propagation to Lagrangian systems
• Uses action extremization for dynamical trajectories
• Enables training without backpropagation through time