Training Coupled Phase Oscillators as a Neuromorphic Platform using Equilibrium Propagation

TL;DR

The paper tackles training neuromorphic systems that operate via physical dynamics, focusing on a network of coupled phase oscillators described by the XY/Kuramoto energy landscape. It applies Equilibrium Propagation (EP) to extract training gradients using two local phases, free and nudge, with the gradient given by $\partial C/\partial \theta_\alpha \approx \tfrac{1}{\beta} (\langle \partial E/\partial \theta_\alpha \rangle^{\rm nudge} - \langle \partial E/\partial \theta_\alpha \rangle^{\rm free})$, and trains with a cost function $C(\phi_{\rm out}, \phi^{\tau})$ designed to avoid unstable fixed points. Numerical demonstrations on XOR and handwritten-digit recognition show that EP can effectively train networks of oscillators, though multistability can complicate learning; this challenge is mitigated by random initialization of hidden/output units and averaging gradients across runs. The results establish coupled phase oscillators as a general-purpose neuromorphic platform and outline realistic pathways for hardware implementations, including laser arrays and other platforms that support phase readout, external driving, and tunable couplings. Overall, the work demonstrates a physics-grounded approach to supervised learning on energy-based neuromorphic systems with practical implications for energy-efficient computation.

Abstract

Given the rapidly growing scale and resource requirements of machine learning applications, the idea of building more efficient learning machines much closer to the laws of physics is an attractive proposition. One central question for identifying promising candidates for such neuromorphic platforms is whether not only inference but also training can exploit the physical dynamics. In this work, we show that it is possible to successfully train a system of coupled phase oscillators - one of the most widely investigated nonlinear dynamical systems with a multitude of physical implementations, comprising laser arrays, coupled mechanical limit cycles, superfluids, and exciton-polaritons. To this end, we apply the approach of equilibrium propagation, which permits to extract training gradients via a physical realization of backpropagation, based only on local interactions. The complex energy landscape of the XY/ Kuramoto model leads to multistability, and we show how to address this challenge. Our study identifies coupled phase oscillators as a new general-purpose neuromorphic platform and opens the door towards future experimental implementations.

Figures (4)

• Figure 1: Equilibrium propagation for coupled phase oscillators. (a) We train a neuromorphic system of coupled phase oscillators (units), here represented by blue circles. In the free phase (left), the input units (orange) are fixed and the system relaxes to its equilibrium via the standard overdamped dynamics of coupled phase oscillators. Output units are highlighted by a blue rim. In the nudge phase (right), the output units are forced weakly towards the desired target output (red arrows) and the the system evolves to a new, slightly shifted equilibrium. The difference between these two equilibria allows to infer the gradients needed to train the system, according to the fundamental rule of equilibrium propagation, see Eq. \ref{['eq:EPgradient']}. (b) Schematic representation of the evolution in the space of oscillator phases $\phi_j$, with open circles depicting initial conditions and full circles depicting equilibria ($\phi^{\tau}$ is the desired target configuration). (c) Possible experimental platforms that can give rise to the dynamics of the XY/ Kuramoto model of coupled phase oscillators include superconducting circuits, nonlinear electrical circuits, exciton-polariton systems, mechanical-oscillator limit cycles, coupled laser arrays and superfluids.
• Figure 2: The problem of multi-stability. (a) A case with only one stable fixed point. The circles refer to the initial states and the arrows refer to the relaxation to the stable fixed points (equilibria; solid dots). The arrows with dashed lines refer the evolution of the equilibria during the training. The dark red dot refers to the target output, assuming that the two phase coordinates shown here both refer to output units, i.e., hidden phases and input phases are suppressed in this depiction. Different colors refer to different iterations during the training. (b) Multistability. Different equilibria can be reached at the same training iteration (same color), starting from different random initial conditions.
• Figure 3: Results of training an XY model on the XOR task. (a) Statistics of the training progress of 5-unit XY networks of all-to-all connectivity (the structure is shown in the upper right corner). Evolution of mean distance function $\langle D \rangle$ (white line, average over 1000 training runs). The density plot indicates the histogram of $\log_{10}D$ over these runs. (b)The dependence of the learning speed on the size $N$. For a given $N$, we randomly initialize the trainable parameters and train them for 1000 iterations, repeating this whole process 100 times. The curves show the evolution of $\overline{\langle D \rangle}$ during the training. (c) Evolution of equilibria during a single training run. For each of the four input configurations, the possible stable fixed points at each training iteration are found by running the system with random initialization for 100 times. For each of these fixed points the state of the output unit (upper row) and one of the hidden units (lower low) are recorded. Output units for all fixed points converge to a unique, deterministic function of the input at the end of training. (d) Dependency of learning speed on the number of random initial configurations, $M_{\rm init}$ For each $N$ and $M_{\rm init}$ we show the evolution of $\overline{\langle D \rangle}$ averaged over 100 training runs. (e) Upper panel: the learning speed of training measured by the average slope of the first 300 steps shown in (d) and its dependence on $M_{\rm init}$. Lower panel: the "physical training speed", measured by the slope divided by $M_{\rm init}$.
• Figure 4: Handwritten-digit recognition in a network of coupled phase oscillators. (a): We consider both all-to-all and layered connectivity networks, where the recognized digit is one-hot encoded in the phase configuration of the 10 output units. (b): Training curves for networks of different sizes with all-to-all connectivity and layer structure. The sizes of the networks are selected such that the number of trainable parameters for them are approximately the same (shown in tab:network_paras) (d): Training evolution of confusion matrices, for different structures and different network sizes.