Mathematical Sciences: Dynamical Systems and Neural Networks

调查者研究几个相互关联的问题 关于神经网络，学习算法和理论 动力系统 首要目标是找出 非线性动力学的特点，特别是动态 吸引子（极限环，混沌吸引子，分形） 显然在生物系统中发现， 学习、识别和检索。 为了测试结果，网 能够识别手写数字的机器将被制造出来。 规范形 投影算法是一种新的网络构造方法， 被利用 以这种方式构建的网络的一个关键特征是， 潜在的动力学是明确同构于任何一类 标准的、易于理解的非线性动力系统。 这 系统是预先选择的，独立于要被 存储和使用的学习算法。 投影算法 包含了某些动力学特征--稳定性，盆地 几何学，和收敛速度--从其他更 依赖于问题的，如预处理，学习规则， 数据重建。动力学中的相关理论问题 系统将进行调查。 神经网络是一种相对简单的计算网络， 节点，通常是输出取决于 非线性地影响其输入。 节点之间的连接可以 它们本身被加权，使得某些输入或输出更 比别人有影响力。 如果一个人把一张网看作是一个动态的 系统，系统的其余点是网络所知道的。" 在此过程中获得的理论知识和实践经验 项目可能会导致更好地理解设计 生物制剂上级性能的基本原则 模式识别、机器人和自适应行为的系统， 以及对记忆、学习和运动控制的更深入了解 在生物网络中。 实际后果出现在 将这些思想应用于模式识别问题， 信号处理、过程控制和通用接口 人与机器之间。

The investigator studies several interrelated questions concerning neural networks, learning algorithms, and theoretical dynamical systems. A primary goal is to discover how certain features of nonlinear dynamics, and in particular the dynamic attractors (limit cycles, chaotic attractors, fractals) apparently found in biological systems, can be useful for pattern learning, recognition, and retrieval. To test the results, nets that recognize handwritten digits will be built. The normal form projection algorithm, a new method of constructing networks, will be used. A key feature of a net built this way is that its underlying dynamics is explicitly isomorphic to any of a class of standard, well-understood nonlinear dynamical systems. This system is chosen in advance, independent of the patterns to be stored and the learning algorithm used. The projection algorithm decouples certain dynamical features -- stability, basin geometry, and rates of convergence -- from others that are more problem-dependent, such as preprocessing, learning rules, and reconstruction of data. Related theoretical issues in dynamical systems will be investigated. A neural net is a network of relatively simple computational nodes, typically summing or damping devices whose output depends nonlinearly on their inputs. The connections between nodes may themselves be weighted so that certain inputs or outputs are more influential than others. If one views a net as a dynamical system, the rest points of the system are what the net "knows." The theoretical knowledge and practical experience gained in this project may lead to greater understanding of the design principles that underlie the superior performance of biological systems in pattern recognition, robotics, and adaptive behavior, and to greater insight into memory, learning, and motor control in biological networks. Practical consequences arise in the applications of these ideas to problems in pattern recognition, signal processing, process control, and general interfaces between people and machines.