论文标题
连续储层计算机的广义同步，嵌入和近似值
Generalised Synchronisations, Embeddings, and Approximations for Continuous Time Reservoir Computers
论文作者
论文摘要
我们建立了条件，在源动力学和储层动力学之间，连续的时间储备计算机（例如泄漏的集成商回声状态网络）承认了广义同步$ f $。我们表明，可以同时存在多个广义同步，并将其连接到多回声 - 状态 - 属性（Multi-Esp）。在线性储层计算机的特殊情况下，我们为广义同步$ f $得出了封闭式表达式。此外，我们确定了$ f $属于$ c^1 $的条件，并且根据$ f $是源系统固定点上的拓扑嵌入的条件。该嵌入结果与Takens的嵌入定理密切相关。 我们还证明，对于随机生成的线性储层系统，固定点的嵌入几乎肯定发生。通过达到嵌入，我们讨论了通用近似定理如何使源系统的未来动态并复制其拓扑特性成为可能。我们通过使用数值方法将Lorenz-63系统的固定点嵌入储层空间中来说明理论。最后，我们表明，如果观察结果受到白噪声的扰动，则将GS保留到Ornstein-Uhlenbeck过程中。
We establish conditions under which a continuous time reservoir computer, such as a leaky integrator echo state network, admits a generalised synchronisation $f$ between between the source dynamics and reservoir dynamics. We show that multiple generalised synchronisations can exist simultaneously, and connect this to the multi-Echo-State-Property (multi-ESP). In the special case of a linear reservoir computer, we derive a closed form expression for the generalised synchronisation $f$. Furthermore, we establish conditions under which $f$ is of class $C^1$, and conditions under which $f$ is a topological embedding on the fixed points of the source system. This embedding result is closely related to Takens' embedding Theorem. We also prove that the embedding of fixed points occurs almost surely for randomly generated linear reservoir systems. With an embedding achieved, we discuss how the universal approximation theorem makes it possible to forecast the future dynamics of the source system and replicate its topological properties. We illustrate the theory by embedding a fixed point of the Lorenz-63 system into the reservoir space using numerical methods. Finally, we show that if the observations are perturbed by white noise, the GS is preserved up to a perturbation by an Ornstein-Uhlenbeck process.
