Training and synchronizing oscillator networks with Equilibrium Propagation

TL;DR

The paper demonstrates that Equilibrium Propagation can train oscillator networks while driving synchronization, even when natural frequencies vary. By formulating an energy $E_\phi$ and a total energy $F_\phi = E_\phi + \beta \mathcal{L}$ and applying two phases (free with $\beta=0$ and nudged with $\beta>0$), it derives gradient updates for couplings $\Omega$ via $\Delta \Omega_{j,k} \propto \frac{1}{\beta}\big(\cos(\phi^\beta_k - \phi^\beta_j) - \cos(\phi^0_k - \phi^0_j)\big)$. The authors validate two oscillator models—the phase-only Kuramoto and a fully synchronized amplitude-phase nonlinear model—on MNIST, achieving $97.77\%$ and $96.85\%$ test accuracies respectively, and show robustness to frequency dispersion up to $5$–$10\%$. These results suggest practical hardware implementations (e.g., spintronic devices) for large-scale, gradient-based learning in oscillator networks, broadening the route to neuromorphic computing with on-chip training capabilities.

Abstract

Oscillator networks represent a promising technology for unconventional computing and artificial intelligence. Thus far, these systems have primarily been demonstrated in small-scale implementations, such as Ising Machines for solving combinatorial problems and associative memories for image recognition, typically trained without state-of-the-art gradient-based algorithms. Scaling up oscillator-based systems requires advanced gradient-based training methods that also ensure robustness against frequency dispersion between individual oscillators. Here, we demonstrate through simulations that the Equilibrium Propagation algorithm enables effective gradient-based training of oscillator networks, facilitating synchronization even when initial oscillator frequencies are significantly dispersed. We specifically investigate two oscillator models: purely phase-coupled oscillators and oscillators coupled via both amplitude and phase interactions. Our results show that these oscillator networks can scale successfully to standard image recognition benchmarks, such as achieving nearly 98\% test accuracy on the MNIST dataset, despite noise introduced by imperfect synchronization. This work thus paves the way for practical hardware implementations of large-scale oscillator networks, such as those based on spintronic devices.