带随机驱动的随机二次型：由噪声实现的亚稳态同步

中文摘要 AI 辅助

我们研究了存在随机布朗驱动时球面上的随机二次型（Random Quadratic Form，RQF）。研究表明，该驱动并未有效改变过程的概率分布，但会影响系统的同步特性：无驱动的RQF因固有对称性呈现部分同步，而引入任意小的驱动会导致长期对称性破缺，进而实现完全同步。本研究聚焦于小驱动区域，恢复了两点过程的多尺度行为：第一阶段，因RQF的对称性，模型收敛至反极性构型；第二阶段，因对称性破缺现象，两个聚类相遇。该模型的灵感来源于连续时间机器学习模型，如神经常微分方程（Neural ODEs）和Transformer的连续时间形式，本研究的结果尤其解释了初始化偏置及其尺度的作用。

英文摘要

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.

发表机构

• Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）

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