Mathematical Sciences: Dynamical Systems and Neural Networks

这个项目结合了动力系统和神经网络的相关工作。P.I.将研究某些类型的差分格式和密切相关的非线性微分方程系统的动力学。这些系统依赖于确定性或随机波动的参数。这些结果将用于评估自适应神经网络几种监督学习方案（包括Widrow-Hoff联想学习和Barto-Anandan强化学习）中权重动态的收敛性。P.I.还将研究由于Amari和Takeuchi造成的波动环境中视网膜发育的神经网络模型。在微分方程的竞争和合作系统中，P.I.将扩展早期的工作，以获得适用于具有不对称固定权重的某些类型的循环神经网络的激活动力学的收敛定理。另一个单独的研究将试图证明R3中的耗散竞争系统，以及R4中的一类有限的这样的系统，可以用结构稳定的系统来逼近，并且满足所有r2的C'闭引理。这将建立结构稳定系统的新类别，并将成为3维中非平凡C2闭引理的第一个证明。

This project combines related work in dynamical systems and neural networks. The P.I. will investigate the dynamics of certain types of difference schemes and closely related systems of nonlinear differential equations. These systems depend on parameters that fluctuate either deterministically or stochastically. These results will be used to evaluate the convergence of the weigh dynamics in several kinds of supervised learning schemes for adaptive neural networks, including Widrow-Hoff associative learning and Barto-Anandan reinforcement learning. The P.I. will also investigate a neural network model of retinal development in fluctuating environments due to Amari and Takeuchi. In competitive and cooperative systems of differential equations, the P.I. will extend earlier work to obtain convergence theorems applicable to the activation dynamics of certain types of recurrent neural networks having asymmetrical fixed weights. A separate study will try to prove that dissipative competitive systems in R3, and a restricted class of such systems in R4, can be approximated by structurally stable ones, and satisfy a C' Closing lemma for all r2. This would establish new classes of structurally stable systems, and would be the first proof of a nontrivial C2 closing lemma in dimensions 3.