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Optical Implementation of Equilibrium Propagation Using Spatial Photonic Ising Machines

arXiv 2606.13454 ↗First posted 11 Jun 2026physics.optics cond-mat.dis-nn cs.ETcited by 0

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Abstract (arXiv)

Equilibrium Propagation offers a compelling alternative to traditional machine learning for training energy-based networks. Here we demonstrate a hybrid optical-digital implementation of EP using a Spatial Photonic Ising Machine (SPIM). The SPIM exploits the gauge transformation method to optically encode both continuous neuron states and rank-1 binary trainable patterns as phase modulations via a spatial light modulator, with inference realized using a finite difference scheme. The experimental system is evaluated on the Wine classification dataset. The potential of this approach, including the use of continuous couplings and structured coupling matrices, is evaluated numerically on the more complex MNIST dataset. Our work provides a concrete pathway toward energy-efficient physical implementations of Equilibrium Propagation.

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References · 25

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
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  3. 2016Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagationcited by 259
  4. 2013Ising formulations of many NP problemscited by 207
  5. 2021Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systemscited by 128
  6. 2020Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Biascited by 64
  7. 2016Deep vs. shallow networks : An approximation theory perspectivecited by 63
  8. 2020Training End-to-End Analog Neural Networks with Equilibrium Propagationcited by 63
  9. 2019Latent Weights Do Not Exist: Rethinking Binarized Neural Network Optimizationcited by 60
  10. 2024Training of Physical Neural Networkscited by 49
  11. 2022Experimentally realized in situ backpropagation for deep learning in nanophotonic neural networkscited by 44
  12. 2019Updates of Equilibrium Prop Match Gradients of Backprop Through Time in an RNN with Static Inputcited by 33
13 more
  1. 2020EqSpike: Spike-driven Equilibrium Propagation for Neuromorphic Implementationscited by 31
  2. 2023Training an Ising Machine with Equilibrium Propagationcited by 29
  3. 2022Beyond backpropagation: bilevel optimization through implicit differentiation and equilibrium propagationcited by 26
  4. 2018Generalization of Equilibrium Propagation to Vector Field Dynamicscited by 23
  5. 2023Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterialcited by 20
  6. 2024Training Coupled Phase Oscillators as a Neuromorphic Platform using Equilibrium Propagationcited by 16
  7. 2022Agnostic Physics-Driven Deep Learningcited by 15
  8. 2021Training Dynamical Binary Neural Networks with Equilibrium Propagationcited by 13
  9. 2024Equilibrium Propagation: the Quantum and the Thermal Casescited by 10
  10. 2023Training neural networks with end-to-end optical backpropagationcited by 7
  11. 2025Equilibrium Propagation for Learning in Lagrangian Dynamical Systemscited by 6
  12. 2025Lagrangian-based Equilibrium Propagation: generalisation to arbitrary boundary conditions & equivalence with Hamiltonian Echo Learningcited by 6
  13. 2023Low-rank combinatorial optimization and statistical learning by spatial photonic Ising machinecited by 4

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