# PotentialEP: Backprop-Free Training of CIFAR-Scale Generative Models via Equilibrium Propagation

arXiv: https://arxiv.org/abs/2610.14409 · HTML: https://mlsanity.com/paper/2610.14409/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 23 of its 35 references that are in the map.
- First posted on arXiv 10 Oct 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 23 of 23)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Energy and Policy Considerations for Deep Learning in NLP](https://mlsanity.com/paper/1906.02243/) (2019)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Power Hungry Processing: Watts Driving the Cost of AI Deployment?](https://mlsanity.com/paper/2311.16863/) (2023)
- [Improved Contrastive Divergence Training of Energy Based Models](https://mlsanity.com/paper/2012.01316/) (2020)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimation](https://mlsanity.com/paper/2106.05527/) (2021)
- [Predictive Coding Approximates Backprop along Arbitrary Computation Graphs](https://mlsanity.com/paper/2006.04182/) (2020)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2006.03824/) (2020)
- [Training End-to-End Analog Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2006.01981/) (2020)
- [Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling](https://mlsanity.com/paper/2504.10612/) (2025)
- [Equilibrium Matching: Generative Modeling with Implicit Energy-Based Models](https://mlsanity.com/paper/2510.02300/) (2025)
- [Spiking Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.17046/) (2023)
- [Guiding Energy-based Models via Contrastive Latent Variables](https://mlsanity.com/paper/2303.03023/) (2023)
- [Dual Propagation: Accelerating Contrastive Hebbian Learning with Dyadic Neurons](https://mlsanity.com/paper/2302.01228/) (2023)
- [Sample as You Infer: Predictive Coding With Langevin Dynamics](https://mlsanity.com/paper/2311.13664/) (2023)
- [Optical Generative Models](https://mlsanity.com/paper/2410.17970/) (2024)
- [ePC: Fast and Deep Predictive Coding in Digital Simulation](https://mlsanity.com/paper/2505.20137/) (2025)
- [Optical Diffusion Models for Image Generation](https://mlsanity.com/paper/2407.10897/) (2024)
- [Local Learning Rules for Out-of-Equilibrium Physical Generative Models](https://mlsanity.com/paper/2506.19136/) (2025)
- [Training a Predictive Coding Network on ImageNet using Equilibrium Propagation](https://mlsanity.com/paper/2606.03584/) (2026)
- [Training a Hopfield Variational Autoencoder with Equilibrium Propagation](https://mlsanity.com/paper/2311.15047/) (2023)

## Abstract (arXiv)

Generating an image with a diffusion or flow-matching model typically involves integrating a velocity field for hundreds of sequential steps. That computation suits a GPU poorly and an analog neuromorphic substrate well, where one fast, low-power physical relaxation could replace the whole sequence. Two things are needed: a generative algorithm whose computation is a relaxation, and a way to train it without backpropagation, whose global gradient transport is hard to realise in analog. Energy Matching supplies the first, sampling by gradient-descent relaxation on a time-independent potential - but obtaining the gradient that descent follows is itself a backward pass, once per integration step. We prove that a predictive coding network with a single linear injection term added to its energy relaxes to a state whose visible layer carries that gradient inherently, with an error that vanishes for small injection strengths. An Energy Matching model trained by backpropagation can therefore be loaded into this network unchanged and sampled by relaxation alone, matching the original model's FID to within 0.04 at CIFAR-10 and ImageNet-32 scale. The network can also be trained without backpropagation: Equilibrium Propagation reproduces backpropagation's parameter updates to a cosine similarity of 0.9998, and training end to end with those updates reaches an FID of 3.64 on unconditional CIFAR-10 against 3.54 for backpropagation. To our knowledge this is the first CIFAR-10 generative model trained with Equilibrium Propagation, and the first at competitive quality by any backpropagation-free rule. This quality does not require group normalisation or attention: an architecture of only convolutions and pointwise nonlinearities, both native to an analog crossbar, reaches an FID of 3.51 when trained by backpropagation.