Mathematical Sciences: Dynamical Systems and Neural Networks

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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.