# Hybridizing Equilibrium Propagation with Ising Machines for Efficient Energy-Based Learning

arXiv: https://arxiv.org/abs/2606.09112 · HTML: https://mlsanity.com/paper/2606.09112/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 47 references that are in the map.
- First posted on arXiv 8 Jun 2026.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [All-Optical Machine Learning Using Diffractive Deep Neural Networks](https://mlsanity.com/paper/1804.08711/) (2018)
- [Training Very Deep Networks](https://mlsanity.com/paper/1507.06228/) (2015)
- [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)
- [Latent Weights Do Not Exist: Rethinking Binarized Neural Network Optimization](https://mlsanity.com/paper/1906.02107/) (2019)
- [Experimentally realized in situ backpropagation for deep learning in nanophotonic neural networks](https://mlsanity.com/paper/2205.08501/) (2022)
- [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)
- [Updates of Equilibrium Prop Match Gradients of Backprop Through Time in an RNN with Static Input](https://mlsanity.com/paper/1905.13633/) (2019)
- [Equivalence of Equilibrium Propagation and Recurrent Backpropagation](https://mlsanity.com/paper/1711.08416/) (2017)
- [Training an Ising Machine with Equilibrium Propagation](https://mlsanity.com/paper/2305.18321/) (2023)
- [Training Coupled Phase Oscillators as a Neuromorphic Platform using Equilibrium Propagation](https://mlsanity.com/paper/2402.08579/) (2024)
- [Training Dynamical Binary Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2103.08953/) (2021)
- [Learning at the Speed of Physics: Equilibrium Propagation on Oscillator Ising Machines](https://mlsanity.com/paper/2510.12934/) (2025)

## Abstract (arXiv)

The rapid evolution of artificial intelligence has led to substantial advances in deep neural networks. Nonetheless, conventional GPU-based training remains highly energy-demanding, motivating the exploration of physical dynamics and compatible energy-based learning schemes, such as equilibrium propagation (EP). EP-based training, however, frequently suffers from convergence to local minima due to phase-space contraction. Here we introduce an Ising-dynamics-inspired equilibrium-propagation framework in which dissipative Hopfield relaxation is replaced by an extended phase-space dynamics with conjugate variables. The resulting training paradigm keeps the local two-phase learning rule of EP while changing the physical route by which neural states reach equilibrium. We show that this dynamics lowers effective energy barriers, accelerates convergence, improves noise robustness, and trains deep convolutional Hopfield networks on MNIST, FashionMNIST, and CIFAR-10 with performance comparable to backpropagation.