Equilibrium Propagation

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Equilibrium propagation studies how to train energy-based neural networks by comparing a network’s free equilibrium with a slightly nudged state, using local changes to estimate gradients instead of backpropagating through every layer. It is explored in predictive-coding and Hopfield networks, as well as in physical circuits, photonic and Ising hardware, oscillators, and neuromorphic controllers; variants address finite nudging and slow relaxation. Evaluations span image classification and generative modeling, while theoretical work examines convergence, conservation laws, and how circuit structure affects learning. A central challenge is making equilibration stable and scalable on physical hardware while achieving reliable performance on deep, complex tasks comparable to backpropagation.

Latest papers

18 papers3 papers in the last four weeks, with none the four weeks before. 0.0% of all new papers.

Generative Replay for Equilibrium Propagation: Toward Edge-Friendly Continual Learning in Hopfield Networks
Oct 27, 2026Victor Cook, Gilles Sassatelli, Thierry Gil +2Replay-Based Continual Learning Generative Replay
LIRMM | ADAC · LIRMM, Univ. Montpellier / CNRS, Montpellier, France · LIRMM +2
PotentialEP: Backprop-Free Training of CIFAR-Scale Generative Models via Equilibrium Propagation
Oct 10, 2026Alex GowerImage Generation Equilibrium Propagation
University of Cambridge & Nokia Bell Labs
An all-transistor physical-learning transformer can be trained to perform in-context learning
Oct 8, 2026Michael C. Abbott, Benjamin B. MachtaTransformer Attention Equilibrium Propagation
Yale Physics & QBio
Lock-in EP: An In-Situ Training Algorithm for Oscillatory Hardware
Oct 5, 2026Sowjanya Tammali, Wilkie Olin-AmmentorpAnalog Computing Equilibrium Propagation
Computer Science, Missouri University of Science and Technology, 500 West 15th Street, Rolla, 65409, MO, USA. · Mathematics & Computer Science Division, Argonne National Laboratory, 9700 S. Cass Avenue, Lemont, 60439, IL, USA.
Conservation laws determine what physical learning remembers
Jul 30, 2026Bijaya DangolEquilibrium Propagation Neural Network Training Dynamics
1Independent researcher
A Conservation Law for Equilibrium Propagation and Coupled Learning
Jun 13, 2026Joshua A. McGinnis, Adam G. Kline, Yoichiro MoriEquilibrium Propagation Neural Network Training Dynamics
Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA · Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, PA, USA · Department of Biology, University of Pennsylvania, Philadelphia, PA, USA
Coercivity and Local Convergence of Physical Learning in Linear Circuits
Jun 13, 2026Joshua A. McGinnis, Xinbo Li, Yoichiro MoriEquilibrium Propagation Optimization Convergence Analysis
Department of Mathematics, University of Pennsylvania, Philadelphia.
Optical Implementation of Equilibrium Propagation Using Spatial Photonic Ising Machines
Jun 11, 2026Dimitri Vanden Abeele, Daniele Veraldi, Davide Pierangeli +2Equilibrium Propagation Energy-Efficient ML
Laboratoire d’Information Quantique C.P. 224, Université Libre de Bruxelles (ULB), Brussels, Belgium · Dipartimento di Fisica, Sapienza Università di Roma, Rome, Italy
Perturbative Contrastive Physical Learning
Jun 8, 2026Kyungeun Kim, Amanuel Anteneh, Israel Klich +2Equilibrium Propagation Contrastive Learning
Department of Mathematics, University of British Columbia, Vancouver, BC Canada · Department of Physics, University of Virginia, 382 McCormick Rd, Charlottesville, VA 22903, USA · Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany +2
Optimizing Energy-based Neural Network Training with Coherent Ising Machine
Jun 8, 2026Chen-Rui Fan, Bo Lu, Zhi-Hong Zhang +3Equilibrium Propagation Neural Network Optimization
School of Artiﬁcial Intelligence, Beijing Normal University, Beijing, 100875, China. · Laboratory for Advanced Computing and Intelligence Engineering, Information Engineering University, Zhengzhou, 450001, China. · China Mobile (Suzhou) Software Technology Company Limited, Suzhou, 215163, China.
Hybridizing Equilibrium Propagation with Ising Machines for Efficient Energy-Based Learning
Jun 8, 2026Chen-Rui Fan, Bo Lu, Xing-Yu Wu +2Equilibrium Propagation Hopfield Networks
School of Artificial Intelligence, Beijing Normal University, Beijing 100875, China · Laboratory for Advanced Computing and Intelligence Engineering, Information Engineering University, Zhengzhou 450001, China · School of Physical Science and Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China
Training a Predictive Coding Network on ImageNet using Equilibrium Propagation
Jun 2, 2026Tugdual Kerjan, Rasmus Høier, Benjamin ScellierEquilibrium Propagation Predictive Coding
Rain AI
Equilibrium Propagation and Hamiltonian Inference in the Diffusive Fitzhugh-Nagumo Model
May 20, 2026Jack KendallEquilibrium Propagation Dynamical Systems
Zyphra · San Francisco, CA
Neuromorphic Reinforcement Learning for Quadruped Locomotion Control on Uneven Terrain
May 10, 2026Zhuangyu Han, Abhronil SenguptaEquilibrium Propagation Terrain-Aware Robot Locomotion
School of Electrical Engineering and Computer Science Penn State University University Park, PA 16802
Dendritic Neural Networks with Equilibrium Propagation
May 1, 2026Yoshimasa KuboEquilibrium Propagation Neural Network Training Dynamics
Department of Computer Science Lakehead University Thunder Bay, Canada
Symmetric Equilibrium Propagation for Thermodynamic Diffusion Training
Apr 26, 2026Aditi DeEquilibrium Propagation Thermodynamic Computing
The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks
Apr 11, 2026Mani Rash AhmadiEquilibrium Propagation Oscillatory Neural Networks
Calibur Labs
Heterogeneous Time Constants Improve Stability in Equilibrium Propagation
Mar 3, 2026Yoshimasa Kubo, Suhani Pragnesh Modi, Smit PatelEquilibrium Propagation Neural Network Training Dynamics
Department of Computer Science, Lakehead University

Oct 27, 2026J/K move · Enter open · S save

Generative Replay for Equilibrium Propagation: Toward Edge-Friendly Continual Learning in Hopfield Networks

Victor Cook, Gilles Sassatelli, Thierry Gil, Mohamed Watfa+1

LIRMM | ADAC · LIRMM, Univ. Montpellier / CNRS, Montpellier, France · LIRMM+2

Replay-Based Continual Learning Generative Replay Catastrophic Forgetting Class-Incremental Learning Hopfield Networks Equilibrium Propagation

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Equilibrium Propagation (EqProp) offers a biologically grounded and hardware-aligned alternative to backpropagation for training Hopfield Networks. However, its behavior in continual learning settings remains largely unexplored. This work confronts catastrophic forgetting in EqProp-trained Hopfield Networks from the inside out, eliminating the need for external data buffers or architectural overhead that standard Experience Replay (ER) and Generative Replay (Gen-Replay) inevitably require. We introduce an Energy-Based Generative Replay strategy (EB-Gen) which exploits the intrinsic generative dynamics of Hopfield Networks, allowing a single model to act as both classifier and replay samples generator. Samples capturing past task knowledge are generated by relaxing a frozen copy of the network, enabling rehearsal solely through internally generated pseudo-data. We evaluate the proposed method in the Class-Incremental setting, using the split versions of the MNIST and CIFAR10 benchmarks and compare it with FineTune, Experience Replay, Generative Replay, Sleep-Like Replay Consolidation (SRC) and Joint training. Our results show that EB-Gen significantly improves stability while preserving plasticity, achieving a 75% and 64% average accuracy on Split MNIST and Split CIFAR10 respectively. These findings show that Equilibrium Propagation can be adapted to alleviate catastrophic forgetting without forgoing the traits which made it compelling in the first place.