Mean-Field Oscillator Ising Machines: Gradient Flows and Classification of Limit Solutions

Abstract

Oscillator Ising Machines (OIMs) have emerged as promising computational architectures for approximating solutions to combinatorial optimization problems. We derive and analyze the mean-field limit of an OIM model and show that it inherits the gradient-flow structure of the finite-dimensional dynamics. We identify conditions under which this mean-field evolution admits an Eulerian formulation as a gradient flow on the Wasserstein space of probability measures, and contrast this with a Lagrangian formulation which is always available. The gradient-flow structure strongly constrains the long-time dynamics and enables a complete classification of limit solutions and their stability in the symmetric case. In particular, all limit solutions are fixed points whose phases cluster into at most four groups, and for almost all parameter values, only binarized fixed points -- those with clusters at $0$ and/or $\pi$ -- can be stable. Since binarized states are exactly those for which a feasible solution to the original problem can be read out, this shows that feasible solutions can almost always be recovered. We provide tight bounds on the parameter thresholds for which fixed points in this binarized family are stable, thereby identifying the threshold for binarization in this model. We also present numerical evidence that the mean-field model correctly predicts behavioral regimes in large random networks, including Erd\H{o}s-R\'enyi networks.