# Equilibrium Propagation for Non-Conservative Systems

arXiv: https://arxiv.org/abs/2602.03670 · HTML: https://mlsanity.com/paper/2602.03670/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 62 references that are in the map.
- First posted on arXiv 3 Feb 2026.

## Builds on (most cited first, 27 of 27)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Towards deep learning with segregated dendrites](https://mlsanity.com/paper/1610.00161/) (2016)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2006.03824/) (2020)
- [Training End-to-End Analog Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2006.01981/) (2020)
- [Memory and information processing in neuromorphic systems](https://mlsanity.com/paper/1506.03264/) (2015)
- [Holomorphic Equilibrium Propagation Computes Exact Gradients Through Finite Size Oscillations](https://mlsanity.com/paper/2209.00530/) (2022)
- [Updates of Equilibrium Prop Match Gradients of Backprop Through Time in an RNN with Static Input](https://mlsanity.com/paper/1905.13633/) (2019)
- [EqSpike: Spike-driven Equilibrium Propagation for Neuromorphic Implementations](https://mlsanity.com/paper/2010.07859/) (2020)
- [Training an Ising Machine with Equilibrium Propagation](https://mlsanity.com/paper/2305.18321/) (2023)
- [Equilibrium Propagation with Continual Weight Updates](https://mlsanity.com/paper/2005.04168/) (2020)
- [Backpropagation at the Infinitesimal Inference Limit of Energy-Based Models: Unifying Predictive Coding, Equilibrium Propagation, and Contrastive Hebbian Learning](https://mlsanity.com/paper/2206.02629/) (2022)
- [Generalization of Equilibrium Propagation to Vector Field Dynamics](https://mlsanity.com/paper/1808.04873/) (2018)
- [Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterial](https://mlsanity.com/paper/2311.00537/) (2023)
- [Training Coupled Phase Oscillators as a Neuromorphic Platform using Equilibrium Propagation](https://mlsanity.com/paper/2402.08579/) (2024)
- [Agnostic Physics-Driven Deep Learning](https://mlsanity.com/paper/2205.15021/) (2022)
- [Dual Propagation: Accelerating Contrastive Hebbian Learning with Dyadic Neurons](https://mlsanity.com/paper/2302.01228/) (2023)
- [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)
- [Improving equilibrium propagation without weight symmetry through Jacobian homeostasis](https://mlsanity.com/paper/2309.02214/) (2023)
- [Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocity](https://mlsanity.com/paper/2406.06482/) (2024)
- [Equilibrium Propagation for Learning in Lagrangian Dynamical Systems](https://mlsanity.com/paper/2505.07363/) (2025)
- [Lagrangian-based Equilibrium Propagation: generalisation to arbitrary boundary conditions & equivalence with Hamiltonian Echo Learning](https://mlsanity.com/paper/2506.06248/) (2025)
- [Near-Equilibrium Propagation training in nonlinear wave systems](https://mlsanity.com/paper/2510.16084/) (2025)
- [A Physical Theory of Backpropagation: Exact Gradients from the Least-Action Principle](https://mlsanity.com/paper/2602.02281/) (2026)
- [Equilibrium Propagation for Complete Directed Neural Networks](https://mlsanity.com/paper/2006.08798/) (2020)
- [Two Tales of Single-Phase Contrastive Hebbian Learning](https://mlsanity.com/paper/2402.08573/) (2024)

## Built on by (most cited first, 3 of 3)

- [Unifying Physical Backpropagation](https://mlsanity.com/paper/2608.11585/) (2026)
- [An all-transistor physical-learning transformer can be trained to perform in-context learning](https://mlsanity.com/paper/2610.12571/) (2026)
- [DeltaNet-PC: Test-Time Memory Without Backpropagation Through Time](https://mlsanity.com/paper/2610.15185/) (2026)

## Abstract (arXiv)

Equilibrium Propagation (EP) is a physics-inspired learning algorithm that uses stationary states of a dynamical system both for inference and learning. In its original formulation it is limited to conservative systems, textiti.e. to dynamics which derive from an energy function. Given their applications, it is important to extend EP to non-conservative systems, textiti.e. systems with non-reciprocal interactions. Previous attempts to generalize EP to such systems failed to compute the exact gradient of the cost function. Here we propose a framework that extends EP to arbitrary non-conservative systems, including feedforward networks. We keep the key property of equilibrium propagation, namely the use of stationary states both for inference and learning. However, we modify the dynamics in the learning phase by a term proportional to the non-reciprocal part of the interaction so as to obtain the exact gradient of the cost function. This algorithm can also be derived using a variational formulation that generates the learning dynamics through an energy function defined over an augmented state space. Numerical experiments show that this algorithm achieves better performance and learns faster than previous proposals.