平均场振子伊辛机：梯度流与极限解的分类

中文摘要 AI 辅助

振子伊辛机（Oscillator Ising Machines, OIMs）已成为求解组合优化问题的有前景的计算架构。我们推导并分析了OIM模型的平均场极限，证明其继承了有限维动力学的梯度流结构。我们确定了该平均场演化可作为概率测度的Wasserstein空间上的梯度流的欧拉公式的条件，并将其与始终可用的拉格朗日公式进行对比。梯度流结构对长时间动力学施加了强约束，使得我们能在对称情形下对极限解及其稳定性进行完整分类。特别地，所有极限解均为不动点，其相位最多聚为4组；对于几乎所有参数值，只有二值化不动点——即相位聚在0和/或π处的不动点——才可能稳定。由于二值化状态恰好是可从中提取原始问题可行解的状态，这表明几乎总能恢复可行解。我们对该二值化族中不动点稳定的参数阈值给出了紧界，从而确定了该模型中二值化的阈值。我们还提供了数值证据，表明该平均场模型能正确预测大型随机网络（包括Erdős-Rényi网络）的行为模式。

英文摘要

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 $π$ -- 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 Erdős-Rényi networks.

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31 pages, 5 figures