揭示了储备池网络中低维主流形的形成机制及其与数据内在结构的关系。
On Dominant Manifolds in Reservoir Computing Networks
- 通过线性连续时间模型，将主模式维度与训练数据信息量直接关联。
- 训练后的主模式可近似原系统柯尔普曼特征函数，揭示其与动态模态分解的联系。
- 适用于研究时间序列建模中的泛化能力，适合动力系统与机器学习交叉研究者。
理解训练如何塑造循环网络动态的几何结构，是时间序列建模的核心问题。本文研究了在时间预测任务中，储备池计算（RC）网络训练过程中低维主导流形的出现机制。针对一个简化的线性连续时间储水库模型，我们直接将主导模式的维度与结构关联到训练数据的内在维度和信息含量。特别地，对于由自治动力系统生成的训练数据，我们发现训练后储水库的主导模式可近似原系统的柯尔普曼特征函数，明确揭示了储备池计算与动态模态分解（DMD）算法之间的联系。通过仿真展示了训练过程中特征值运动生成主导流形的过程，并讨论了基于切向动力学与微分p-支配性的非线性储备池推广方法。
原文摘要 · Abstract (English)
Understanding how training shapes the geometry of recurrent network dynamics is a central problem in time-series modeling. We study the emergence of low-dimensional dominant manifolds in the training of Reservoir Computing (RC) networks for temporal forecasting tasks. For a simplified linear and continuous-time reservoir model, we link the dimensionality and structure of the dominant modes directly to the intrinsic dimensionality and information content of the training data. In particular, for training data generated by an autonomous dynamical system, we relate the dominant modes of the trained reservoir to approximations of the Koopman eigenfunctions of the original system, illuminating an explicit connection between reservoir computing and the Dynamic Mode Decomposition algorithm. We illustrate the eigenvalue motion that generates the dominant manifolds during training in simulation, and discuss generalization to nonlinear RC via tangent dynamics and differential p-dominance.
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