# Equilibrium Propagation with Continual Weight Updates

arXiv: https://arxiv.org/abs/2005.04168 · HTML: https://mlsanity.com/paper/2005.04168/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2020.

## Built on by (most cited first, 26 of 26)

- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [SoftHebb: Bayesian Inference in Unsupervised Hebbian Soft Winner-Take-All Networks](https://mlsanity.com/paper/2107.05747/) (2021)
- [Backpropagation at the Infinitesimal Inference Limit of Energy-Based Models: Unifying Predictive Coding, Equilibrium Propagation, and Contrastive Hebbian Learning](https://mlsanity.com/paper/2206.02629/) (2022)
- [A deep learning theory for neural networks grounded in physics](https://mlsanity.com/paper/2103.09985/) (2021)
- [A Stable, Fast, and Fully Automatic Learning Algorithm for Predictive Coding Networks](https://mlsanity.com/paper/2212.00720/) (2022)
- [Bottom-up and top-down approaches for the design of neuromorphic processing systems: Tradeoffs and synergies between natural and artificial intelligence](https://mlsanity.com/paper/2106.01288/) (2021)
- [Agnostic Physics-Driven Deep Learning](https://mlsanity.com/paper/2205.15021/) (2022)
- [Dual Propagation: Accelerating Contrastive Hebbian Learning with Dyadic Neurons](https://mlsanity.com/paper/2302.01228/) (2023)
- [Training Dynamical Binary Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2103.08953/) (2021)
- [Equilibrium Propagation: the Quantum and the Thermal Cases](https://mlsanity.com/paper/2405.08467/) (2024)
- [Constrained Parameter Inference as a Principle for Learning](https://mlsanity.com/paper/2203.13203/) (2022)
- [Improving equilibrium propagation without weight symmetry through Jacobian homeostasis](https://mlsanity.com/paper/2309.02214/) (2023)
- [Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocity](https://mlsanity.com/paper/2406.06482/) (2024)
- [Towards the Training of Deeper Predictive Coding Neural Networks](https://mlsanity.com/paper/2506.23800/) (2025)
- [Equilibrium Propagation for Learning in Lagrangian Dynamical Systems](https://mlsanity.com/paper/2505.07363/) (2025)
- [Equilibrium Aggregation: Encoding Sets via Optimization](https://mlsanity.com/paper/2202.12795/) (2022)
- [Analog In-memory Training on General Non-ideal Resistive Elements: The Impact of Response Functions](https://mlsanity.com/paper/2502.06309/) (2025)
- [Flexible Phase Dynamics for Bio-Plausible Contrastive Learning](https://mlsanity.com/paper/2302.12431/) (2023)
- [Equilibrium Propagation for Non-Conservative Systems](https://mlsanity.com/paper/2602.03670/) (2026)
- [Progressive Fusion for Multimodal Integration](https://mlsanity.com/paper/2209.00302/) (2022)
- [Toward Practical Equilibrium Propagation: Brain-inspired Recurrent Neural Network with Feedback Regulation and Residual Connections](https://mlsanity.com/paper/2508.11659/) (2025)
- [Unsupervised End-to-End Training with a Self-Defined Target](https://mlsanity.com/paper/2403.12116/) (2024)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2101.05536/) (2021)
- [Convergence and scaling of Boolean-weight optimization for hardware reservoirs](https://mlsanity.com/paper/2305.07908/) (2023)
- [Calibrating Neural Networks' parameters through Optimal Contraction in a Prediction Problem](https://mlsanity.com/paper/2406.10703/) (2024)
- [Scalable Equilibrium Propagation via Intermediate Error Signals for Deep Convolutional CRNNs](https://mlsanity.com/paper/2508.15989/) (2025)

## Abstract (arXiv)

Equilibrium Propagation (EP) is a learning algorithm that bridges Machine Learning and Neuroscience, by computing gradients closely matching those of Backpropagation Through Time (BPTT), but with a learning rule local in space. Given an input x and associated target y, EP proceeds in two phases: in the first phase neurons evolve freely towards a first steady state; in the second phase output neurons are nudged towards y until they reach a second steady state. However, in existing implementations of EP, the learning rule is not local in time: the weight update is performed after the dynamics of the second phase have converged and requires information of the first phase that is no longer available physically. In this work, we propose a version of EP named Continual Equilibrium Propagation (C-EP) where neuron and synapse dynamics occur simultaneously throughout the second phase, so that the weight update becomes local in time. Such a learning rule local both in space and time opens the possibility of an extremely energy efficient hardware implementation of EP. We prove theoretically that, provided the learning rates are sufficiently small, at each time step of the second phase the dynamics of neurons and synapses follow the gradients of the loss given by BPTT (Theorem 1). We demonstrate training with C-EP on MNIST and generalize C-EP to neural networks where neurons are connected by asymmetric connections. We show through experiments that the more the network updates follows the gradients of BPTT, the best it performs in terms of training. These results bring EP a step closer to biology by better complying with hardware constraints while maintaining its intimate link with backpropagation.