Reservoir computing is a machine learning framework that has been shown to be able to replicate the chaotic attractor, including the fractal dimension and the entire Lyapunov spectrum, of the dynamical system on which it is trained. We quantitatively relate the generalized synchronization dynamics of a driven reservoir during the training stage to the performance of the trained reservoir computer at the attractor reconstruction task. We show that, in order to obtain successful attractor reconstruction and Lyapunov spectrum estimation, the largest conditional Lyapunov exponent of the driven reservoir must be significantly more negative than the most negative Lyapunov exponent of the target system. We also find that the maximal conditional Lyapunov exponent of the reservoir depends strongly on the spectral radius of the reservoir adjacency matrix, and therefore, for attractor reconstruction and Lyapunov spectrum estimation, small spectral radius reservoir computers perform better in general. Our arguments are supported by numerical examples on well-known chaotic systems.

翻译：储层计算是一种机器学习框架，已被证明能够复现其所训练动力系统的混沌吸引子（包括分形维数和完整李雅普诺夫指数谱）。我们定量分析了训练阶段驱动储层的广义同步动力学与训练后储层计算机在吸引子重构任务中的表现之间的关系。研究表明，为成功实现吸引子重构和李雅普诺夫指数谱估计，驱动储层的最大条件李雅普诺夫指数必须显著小于目标系统的最小李雅普诺夫指数。我们还发现，储层最大条件李雅普诺夫指数强烈依赖于储层邻接矩阵的谱半径，因此，在吸引子重构和李雅普诺夫指数谱估计中，小谱半径的储层计算机通常表现更优。我们的论点通过经典混沌系统的数值算例得到了验证。