噪声让随机二次型系统从部分同步变为完全同步。
Random Quadratic Form with random forcing: Metastable synchronization by noise
- 小噪声使系统在多尺度下经历对极配置与聚类合并
- 即使微弱扰动也能引发长期对称性破缺和完全同步
- 对神经ODE与Transformer的初始化机制有解释作用
我们研究了在球面上受随机布朗力驱动的随机二次型（RQF）系统。结果表明，该力虽不改变过程的分布规律，却显著影响系统的同步特性。无外力时，由于内在对称性，系统呈现部分同步；而引入任意小的随机力后，系统在长期中出现对称性破缺，实现完全同步。本文聚焦小力扰动情形，揭示了两点过程的多尺度行为：第一阶段，系统因对称性演化为反极配置；第二阶段，对称性破缺导致两簇汇聚。该模型源于连续时间机器学习模型，如神经微分方程（Neural ODEs）和Transformer的连续形式，可解释偏置项及其初始化尺度的作用。
原文摘要 · Abstract (English)
We study the Random Quadratic Form (RQF) on a sphere in the presence of random Brownian forcing. We show that the forcing does not effectively change the law of the process but affects the synchronization properties of the system. While the RQF without forcing exhibits partial synchronization due to the intrinsic symmetries, the introduction of an arbitrarily small forcing results in long-term symmetry breaking and leads to full synchronization. In this work we focus on the small forcing regime and recover the multiscale behavior of the two-point process. We show that in the first stage the model converges to an anti-polar configuration due to the symmetries of the RQF and in the second stage the two clusters meet due to the symmetry breaking phenomenon. The model is motivated by continuous-time machine learning models such as Neural ODEs and continuous-time formulations of transformers. In particular, the results of this work explain the role of the bias and the scale of its initialization.
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