用四体耦合相振子网络实现高密度记忆存储,突破传统模型极限。
Higher-Order Kuramoto Oscillator Network for Dense Associative Memory
- 引入四体相位耦合的赫布型振子模型,增强记忆关联能力。
- 当四体耦合为两体耦合的3倍时出现连续到不连续的跃迁转变。
- 噪声下记忆保持时间呈指数增长,适合高鲁棒性存储系统研究。
当相振子网络包含超越经典库朗莫特模型的成对相互作用的真实多体耦合时,可作为高密度关联记忆。本文提出一种广义赫布库朗莫特模型,结合传统的二体一阶傅里叶谐波耦合与真实的四体相位相互作用,灵感来自密集霍普菲尔德记忆理论。对于带有朗之万噪声的同质振子,平衡态均场理论在四体耦合为两体耦合三倍时出现三临界点,此时连续检索开始转为不连续、具有滞后性的相变。我们还使用奥特-安东森方法分析了具有洛伦兹频率分布的确定性模型,发现当四体耦合为两体耦合六倍时,相变从超临界变为亚临界;两种描述仅在线性失稳阈值处一致。在四体主导的平衡区域,存储模式与无序状态共存。我们确定了双稳态区域及双向转换的自由能势垒,并证明噪声诱导的记忆丢失时间随振子数呈指数增长,速率由相关势垒决定。在有限记忆负载下,采用表观腔信号-噪声闭合模型模拟模式串扰,仿真测得有限尺寸下的强线索恢复与保持阈值。在模拟尺寸范围内,幂律拟合的指数大于1,部分四体主导情况中最大指数出现在纯四体零温数据;这些拟合并非渐近容量定律。
原文摘要 · Abstract (English)
Networks of phase oscillators can serve as dense associative memories when they incorporate genuine many-body coupling beyond the classical Kuramoto model's pairwise interaction. Here we introduce a generalized Hebbian Kuramoto model that combines a conventional two-body, first-Fourier-harmonic coupling with a genuine four-body phase interaction, inspired by dense Hopfield memory theory. For identical oscillators with Langevin noise, equilibrium mean-field theory yields a phase diagram with a tricritical point when the four-body coupling is three times the pairwise coupling, where continuous retrieval onset gives way to a discontinuous, hysteretic transition. We separately analyze a deterministic model with Lorentzian frequency disorder using the Ott--Antonsen ansatz. In that model the onset changes from supercritical to subcritical when the four-body coupling is six times the pairwise coupling, and the two descriptions agree only at the linear instability threshold. In the four-body-dominated equilibrium regime, stored-pattern and incoherent states coexist. We determine the bistable region and the free-energy barriers for transitions in both directions, and show that the noise-induced memory-loss time grows exponentially with the number of oscillators, with a rate set by the relevant barrier. At finite memory load, a phenomenological cavity signal--to--noise closure models pattern crosstalk, while simulations measure finite-size strong-cue recovery and retention thresholds. Nominal power-law fits over the simulated size range have exponents above one in several four-body-dominated cases, with the largest fitted exponent in the purely four-body zero-temperature data; these fits are not asymptotic capacity laws.
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