# Near-Equilibrium Propagation training in nonlinear wave systems

arXiv: https://arxiv.org/abs/2510.16084 · HTML: https://mlsanity.com/paper/2510.16084/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 36 references that are in the map.
- First posted on arXiv 17 Oct 2025.

## Builds on (most cited first, 18 of 18)

- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Photonics for artificial intelligence and neuromorphic computing](https://mlsanity.com/paper/2011.00111/) (2020)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2006.03824/) (2020)
- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [Experimentally realized in situ backpropagation for deep learning in nanophotonic neural networks](https://mlsanity.com/paper/2205.08501/) (2022)
- [Wave Physics as an Analog Recurrent Neural Network](https://mlsanity.com/paper/1904.12831/) (2019)
- [Backpropagation-free Training of Deep Physical Neural Networks](https://mlsanity.com/paper/2304.11042/) (2023)
- [The physics of optical computing](https://mlsanity.com/paper/2308.00088/) (2023)
- [Equivalence of Equilibrium Propagation and Recurrent Backpropagation](https://mlsanity.com/paper/1711.08416/) (2017)
- [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)
- [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)
- [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)
- [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)

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

- [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)
- [Unifying Physical Backpropagation](https://mlsanity.com/paper/2608.11585/) (2026)

## Abstract (arXiv)

Backpropagation learning algorithm, the workhorse of modern artificial intelligence, is notoriously difficult to implement in physical neural networks. Equilibrium Propagation (EP) is an alternative with comparable efficiency and strong potential for in-situ training. We extend EP learning to both discrete and continuous complex-valued wave systems. In contrast to previous EP implementations, our scheme is valid in the weakly dissipative regime, and readily applicable to a wide range of physical settings, even without well defined nodes, where trainable inter-node connections can be replaced by trainable local potential. We test the method in driven-dissipative exciton-polariton condensates governed by generalized Gross-Pitaevskii dynamics. Numerical studies on standard benchmarks, including a simple logical task and handwritten-digit recognition, demonstrate stable convergence, establishing a practical route to in-situ learning in physical systems in which system control is restricted to local parameters.