# A Conservation Law for Equilibrium Propagation and Coupled Learning

arXiv: https://arxiv.org/abs/2606.15444 · HTML: https://mlsanity.com/paper/2606.15444/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 20 references that are in the map.
- First posted on arXiv 13 Jun 2026.

## Builds on (most cited first, 5 of 5)

- [Energy and Policy Considerations for Deep Learning in NLP](https://mlsanity.com/paper/1906.02243/) (2019)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Power Hungry Processing: Watts Driving the Cost of AI Deployment?](https://mlsanity.com/paper/2311.16863/) (2023)
- [Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterial](https://mlsanity.com/paper/2311.00537/) (2023)
- [A universal approximation theorem for nonlinear resistive networks](https://mlsanity.com/paper/2312.15063/) (2023)

## Built on by (most cited first, 2 of 2)

- [Untrainable elements determine what physical learning remembers](https://mlsanity.com/paper/2608.00097/) (2026)
- [Reciprocity Separates Gradient Flow from Rotation in Conservative Physical Learning](https://mlsanity.com/paper/2608.30778/) (2026)

## Abstract (arXiv)

In this paper we show that the physical learning methods known as coupled learning (CL) and equilibrium propagation (EP) conserve a mass-like quantity in the trainable parameters in the continuous-time, small-nudging limit. We prove that this conservation holds in a broad range of physically relevant settings. We then show that the conservation law constrains the training dynamics in a way that makes convergence reliable in important settings for linear circuits. We conclude by discussing some practical implications of this conservation law.