Mathematical Sciences: Dynamical Systems and Neural Networks

该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组的动力学。这些系统依赖于确定性或随机性波动的参数。这些结果将被用来评估几种自适应神经网络监督学习方案的权值动态的收敛，包括Widrow-Hoff联想学习和Barto-Ananda强化学习。P.I.还将研究由于Amari和Takeuchi的影响，视网膜在波动环境中发育的神经网络模型。在竞争与合作的微分方程组中，P.I.将推广以前的工作，得到适用于某些类型的具有不对称固定权的递归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中的耗散竞争系统和R4中的一类受限的耗散竞争系统可以被结构稳定的系统逼近，并且满足所有R2的C‘闭引理。这将建立一类新的结构稳定的系统，并将是3维空间中非平凡C2闭合引理的第一个证明。

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.