This work is devoted to the modeling and investigation of the architecture design for the delayed recurrent neural network, based on the delayed differential equations. The usage of discrete and distributed delays makes it possible to model the calculation of the next states using internal memory, which corresponds to the artificial recurrent neural network architecture used in the field of deep learning. The problem of exponential stability of the models of recurrent neural networks with multiple discrete and distributed delays is considered. For this purpose, the direct method of stability research and the gradient descent method is used. The methods are used consequentially. Firstly we use the direct method in order to construct stability conditions (resulting in an exponential estimate), which include the tuple of positive definite matrices. Then we apply the optimization technique for these stability conditions (or of exponential estimate) with the help of a generalized gradient method with respect to this tuple of matrices. The exponential estimates are constructed on the basis of the Lyapunov–Krasovskii functional. An optimization method of improving estimates is offered, which is based on the notion of the generalized gradient of the convex function of the tuple of positive definite matrices. The search for the optimal exponential estimate is reduced to finding the saddle point of the Lagrange function.

中文翻译：

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构建延迟循环神经网络指数估计的优化技术

这项工作致力于基于延迟微分方程对延迟递归神经网络的架构设计进行建模和研究。离散和分布式延迟的使用使得可以使用内部存储器对下一个状态的计算进行建模，这对应于深度学习领域中使用的人工循环神经网络架构。考虑了具有多个离散和分布式延迟的递归神经网络模型的指数稳定性问题。为此，使用了稳定性研究的直接方法和梯度下降法。这些方法是相应地使用的。首先，我们使用直接方法来构建稳定性条件（导致指数估计），其中包括正定矩阵的元组。然后，我们在关于这个矩阵元组的广义梯度方法的帮助下，对这些稳定性条件（或指数估计）应用优化技术。指数估计是在 Lyapunov-Krasovskii 泛函的基础上构建的。提出了一种改进估计的优化方法，该方法基于正定矩阵元组凸函数的广义梯度概念。对最优指数估计的搜索简化为寻找拉格朗日函数的鞍点。提出了一种改进估计的优化方法，该方法基于正定矩阵元组凸函数的广义梯度概念。对最优指数估计的搜索简化为寻找拉格朗日函数的鞍点。提出了一种改进估计的优化方法，该方法基于正定矩阵元组凸函数的广义梯度概念。对最优指数估计的搜索简化为寻找拉格朗日函数的鞍点。