# Equilibrium Propagation for Learning in Lagrangian Dynamical Systems

arXiv: https://arxiv.org/abs/2505.07363 · HTML: https://mlsanity.com/paper/2505.07363/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 22 references that are in the map.
- First posted on arXiv 12 May 2025.

## Builds on (most cited first, 15 of 15)

- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2006.03824/) (2020)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Holomorphic Equilibrium Propagation Computes Exact Gradients Through Finite Size Oscillations](https://mlsanity.com/paper/2209.00530/) (2022)
- [Energy-based learning algorithms for analog computing: a comparative study](https://mlsanity.com/paper/2312.15103/) (2023)
- [EqSpike: Spike-driven Equilibrium Propagation for Neuromorphic Implementations](https://mlsanity.com/paper/2010.07859/) (2020)
- [Self-learning Machines based on Hamiltonian Echo Backpropagation](https://mlsanity.com/paper/2103.04992/) (2021)
- [Equilibrium Propagation with Continual Weight Updates](https://mlsanity.com/paper/2005.04168/) (2020)
- [Generalization of Equilibrium Propagation to Vector Field Dynamics](https://mlsanity.com/paper/1808.04873/) (2018)
- [A deep learning theory for neural networks grounded in physics](https://mlsanity.com/paper/2103.09985/) (2021)
- [Comprehensive Review of Neural Differential Equations for Time Series Analysis](https://mlsanity.com/paper/2502.09885/) (2025)
- [Equilibrium Propagation: the Quantum and the Thermal Cases](https://mlsanity.com/paper/2405.08467/) (2024)
- [Quantum Equilibrium Propagation: Gradient-Descent Training of Quantum Systems](https://mlsanity.com/paper/2406.00879/) (2024)
- [Trainable and Dynamic Computing: Error Backpropagation through Physical Media](https://mlsanity.com/paper/1407.6637/) (2014)
- [Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocity](https://mlsanity.com/paper/2406.06482/) (2024)

## Built on by (most cited first, 6 of 6)

- [Near-Equilibrium Propagation training in nonlinear wave systems](https://mlsanity.com/paper/2510.16084/) (2025)
- [Equilibrium Propagation for Non-Conservative Systems](https://mlsanity.com/paper/2602.03670/) (2026)
- [Training a Predictive Coding Network on ImageNet using Equilibrium Propagation](https://mlsanity.com/paper/2606.03584/) (2026)
- [Efficient Agentic Reasoning Through Self-Regulated Simulative Planning](https://mlsanity.com/paper/2605.22138/) (2026)
- [Optical Implementation of Equilibrium Propagation Using Spatial Photonic Ising Machines](https://mlsanity.com/paper/2606.13454/) (2026)
- [Prospective Coding Improves Learning in Deep Continuous-Time Recurrent Networks](https://mlsanity.com/paper/2609.04134/) (2026)

## Abstract (arXiv)

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.