Creation and control of Fractal spatiotemporal hierarchical structure

将系统视为封闭的，即自治系统。另一种观点认为该系统是开放的，即非自治系统。前一种方法假设外部环境由组成的子系统组成。与此相关的是，许多研究者对混沌现象很感兴趣。因此，对于自治系统来说，混沌被确立为一个重要的研究领域，这可能是发展对自然界复杂系统的理解的关键。另一方面，对非自治系统的研究还处于起步阶段，尚未建立明确的理论框架。我们认为，迄今在这一领域几乎没有进行研究的主要原因是难以审查系统之间的相互作用。因此，我们必须尝试确定如何从动力系统的角度来描述这种相互作用。由时间扩展…随机激励的连续动力系统理论提供了更多的RNAL输入。该理论认为，连续时间递归神经网络(RNN)的动力学一般由超圆柱形相空间中一组具有类分形结构的连续轨迹来描述。我们将这种动力学称为分形化转变。讨论了三种类型的数值实验，以考察学习过程和噪声效应对分形转变的影响。首先，为了分析RNN如何学习期望的输入{输出转换，详细分析了一个具有单一状态的简单示例。在学习过程中，观察到一种类似于Cantor集的分形结构。这一发现为神经网络的学习提供了启示，即神经网络的学习是一个调整分维的过程。其次，研究了输入噪声对分形结构的影响。结果表明，噪声破坏了小尺度的层次结构。第三，利用一个具有20个状态的网络，我们证明了分形转变是开关输入驱动的神经网络的普遍特征。较少

views the system as being closed, I.e. an autonomous system. Another views the system as being open, I.e. a non-autonomous system. The former approach assumes the external environment to be composed of constituent subsystems. In relation to this approach, many researchers have been interested in chaotic phenomena. Consequently, for the autonomous system, chaos has been established as a prominent research area that might be key to developing an understanding of complex systems in nature. On the other hand, the investigation into non-autonomous systems is still in its initial phase, and no clear theoretical framework has yet been established. We believe that the main reason that little research has thus far been done in this area is the difficulty in examining the interaction between systems. Therefore, we must attempt to determine how we can describe this interaction from a dynamical systems point of view. A theory for continuous dynamical systems stochastically excited by temporal exte … More rnal inputs has been presented. The theory suggests that the dynamics of continuous-time recurrent neural networks (RNNs) is generally characterized by a set of continuous trajectories with a fractal-like structure in hyper-cylindrical phase space. We refer to this dynamics as the fractal transition. Three types of numerical experiments are discussed in order to investigate the learning process and noise effects in terms of the fractal transition. First, to analyze how an RNN learns desired input {output transformations, a simple example with a single state was examined in detail. A fractal structure similar to a Cantor set was clearly observed in the learning process. This finding sheds light on the learning of RNNs, I.e. it suggests that the learning is a process of adjusting the fractal dimension. Second, input noise effects on the fractal structure were investigated. The results show that small-scale hierarchical structures are broken by noise. Third, using a network with twenty states, we show that fractal transition is a universal characteristic of RNNs driven by switching inputs. Less