FlowEqProp: Training Flow Matching Generative Models with Gradient Equilibrium Propagation

TL;DR

This paper introduces Gradient Equilibrium Propagation (GradEP), extending equilibrium propagation to train flow-based generative models using local measurements, suitable for neuromorphic hardware, demonstrated on digit recognition.

Contribution

The paper presents GradEP, a novel method for training flow matching generative models with local equilibrium measurements, enabling neuromorphic-compatible training without backpropagation.

Findings

Successfully trained a flow-based generative model on handwritten digits.

Generated recognizable digit samples with stable training dynamics.

Extended generation produces sharper samples through additional inference.

Abstract

We introduce Gradient Equilibrium Propagation (GradEP), a mechanism that extends Equilibrium Propagation (EP) to train energy gradients rather than energy minima, enabling EP to be applied to tasks where the learning objective depends on the velocity field of a convergent dynamical system. Instead of fixing the input during dynamics as in standard EP, GradEP introduces a spring potential that allows all units, including the visible units, to evolve, encoding the learned velocity in the equilibrium displacement. The spring and resulting nudge terms are both purely quadratic, preserving EP's hardware plausibility for neuromorphic implementation. As a first demonstration, we apply GradEP to flow matching for generative modelling - an approach we call FlowEqProp - training a two-hidden-layer MLP (24,896 parameters) on the Optical Recognition of Handwritten Digits dataset using only local equilibrium measurements and no backpropagation. The model generates recognisable digit samples across all ten classes with stable training dynamics. We further show that the time-independent energy landscape enables extended generation beyond the training horizon, producing sharper samples through additional inference-time computation - a property that maps naturally onto neuromorphic hardware, where longer relaxation yields higher-quality outputs. To our knowledge, this is the first demonstration of EP training a flow-based generative model.