Research

Equilibrium Propagation on analog hardware

Equilibrium Propagation (EP) trains a physical system by letting its own dynamics settle to equilibrium, nudging the output towards the target, and reading the gradient out of the difference — no backpropagation graph required. I developed an EP learning algorithm for Oscillator Ising Machines (OIMs), analog networks of coupled oscillators that compute by synchronising. The approach trains both multilayer-perceptron and convolutional architectures directly in the physics of the machine, reaching state-of-the-art neuromorphic accuracy on MNIST and FashionMNIST while remaining resilient to the noise, parameter drift, and quantisation constraints of real analog hardware. A hardware implementation is in its early stages with Nokia and TU Eindhoven.

Gradient EP and generative models

Standard EP learns by shaping energy minima. Gradient Equilibrium Propagation (GradEP) is a general mechanism I developed that extends the framework to train the energy gradients themselves, opening up a different class of models: those defined by vector fields rather than fixed points. It replaces EP’s hard input clamp with a spring potential, so that visible units evolve too and their equilibrium displacement encodes the learned velocity — a purely quadratic modification that keeps the algorithm hardware-plausible. GradEP applies wherever the objective depends on an energy gradient: flow matching, score matching, energy-based generation.

The first application, FlowEqProp, is the first flow matching generative model trained with Equilibrium Propagation (Best Paper Award, ICONS ‘26). Because the learned energy landscape is time-independent, generation can be integrated past the training horizon: the system simply settles deeper, producing sharper samples from additional inference-time compute — which on neuromorphic hardware means nothing more than letting the physics run longer. Ongoing work combines EP with predictive coding networks and scales it to more expressive energy-based architectures, including modern Hopfield networks and energy transformers.

Rugged energy landscapes and glassy dynamics

The third thread asks why gradient descent on rugged landscapes behaves the way it does. Using the Rubik’s Cube as a model statistical-physics system, we uncovered a saddles-to-minima topological crossover in the connectivity of critical points: a structural transition in the landscape that controls the onset of glassy dynamics, and which is suppressed when swap moves are allowed. The Cube turns out to be a surprisingly clean laboratory for questions that matter equally for spin glasses and for optimisation in machine learning.