Mathematical Sciences: A Neural Network, Modeled by a Nonlinear Dynamical System

这个项目研究了一个连续神经网络的性质，它已经被证明成功地解决了模式分类中的某些问题。该网络的数学模型仅使用最基本的假设导出；即，神经网络是相互作用的元素的集合，每个元素都受到其他元素的某个子集的积极或负面影响，从而使系统作为一个整体能够随着时间的推移调整其行为。由此得到的数学模型是一个自治的常微分方程组，因此可以用已知的解析和数值技术进行分析。对于适当的参数取值，可以证明一个N单元网络将有N个稳定吸引子。该项目的主要目标之一是确定这些吸引子的吸引盆地，并描述它们对各种系统参数的依赖关系。然后，该信息将被用来确定以这些N单元网络为组件的神经网络的属性。神经网络目前正在被应用于某些类型的问题，这些问题没有屈服于传统的人工智能技术，例如语音识别和计算机视觉。到目前为止，对于哪种神经网络结构工作得最好，还没有达成一致意见。由具有不同架构的互连组件组成的混合系统可能是一种解决方案。正是本着这种精神，人们正在研究这里提出的网络。它具有一些其他网络没有的有用特性；例如，它自动识别输入矢量中最不同于所有其他分量的分量，无论它是更小还是更大。作为少数几个权重随活动水平而连续变化的模型之一，它在这些权重中具有内置的短期记忆能力。需要将该模型与当前使用的其他神经网络进行仔细的比较，以确定它是否具有使其在特定应用中显著更好的特性。

This project studies the properties of a continuous neural network that has already been shown to successfully solve certain problems in pattern classification. The mathematical model for this network was derived using only the most basic assumptions; namely, that a neural network is a collection of interacting elements, each one affected positively or negatively by some subset of the others, and such that the system as a whole is capable of adapting its behavior over time. The resulting mathematical model is an autonomous system of ordinary differential equations, and can therefore be analyzed by known analytic and numerical techniques. For appropriate values of the parameters it can be shown that an N-cell network will have N stable attractors. One of the major goals of this project is the determination of the attractive basins of these attractors and a description of their dependence on the various system parameters. This information will then be used to determine the properties of a neural network having these N-cell networks as components. Neural networks are currently being applied to certain kinds of problems that have not yielded to traditional techniques of artificial intelligence, for example speech recognition and computer vision. As yet, there is no general agreement on which neural network architectures work best. It is possible that hybrid systems, consisting of interconnected components with different architectures, may be one solution. It is in this spirit that the network proposed here is being studied. It has some useful properties not possessed by other networks; for example, it automatically identifies the component of an input vector that is most distinct from all of the others, whether it is smaller or larger. As one of the few models in which the weights vary continuously in time as a function of the activity levels, it has in these weights a built-in short-term memory capability. A careful comparison of this model with other neural networks in current use needs to be made, to determine whether it has properties that make it significantly better in particular applications.