交集上变分不等式的神经网络模型及应用研究

Using the theory of dynamical system and differential inclusion, this project will design the effective neural-network models for solving variational inequalities and their constrained problems defined on the intersection of the closed sets according to the inherent properties of the considered problems, respectively. These models include continuous-time neural networks, the ones with delays, discontinuous ones for the set-valued mapping and so on. Meanwhile, a suitable energy function is defined to analyze the dynamical behavior for each model by using optimization methods, and some easily checked stability conditions will be provided such that each model has very simpler structure, smaller size, better convergence performance and no requirement of computing directly the projection on the intersection of the closed sets. Moreover, based on the forms and the properties of the optimization problems in the fields of image processing, statistics, machine learning and so on, this project will also design several kinds of the neural-network models with simple structure, small size and no penalty parameter for them, and construct the energy functions to study their stability and convergence by using the suitable transformation and the idea to develop the models for variational inequality defined on the intersection of the closed sets. The significance of this project is that the neural networks, which is more suitable for parallel implementation in simple hardware units, can not only solve practical problems in real time, but also obtain better solution for the original problem with less computation cost by using its discrete version and the theoretical foundations to design and understand numerically stable algorithms.

利用动力系统和微分包含, 本项目将根据问题的内在特性, 分类设计能求解定义于闭集的交集上的变分不等式及其约束问题的神经网络模型, 包括连续时间神经网络、具有时滞的神经网络和适合于集值映射的不连续神经网络模型等；利用优化技巧构造恰当的能量函数, 分析模型的动态行为, 并给出易于验证的稳定性条件, 使新模型结构更简单、规模更小、收敛性能更理想且不需要计算交集上的投影. 其次, 针对图像处理和统计与机器学习等领域中出现的许多优化问题, 将基于问题的形式和特点, 通过适当地转化并应用设计求解交集上变分不等式模型的思路, 分类设计能求解它们的结构简单、规模小且不含罚参数的各种神经网络；定义恰当的能量函数, 研究新模型的稳定性和收敛性. 其意义在于适合硬件实现的神经网络不仅能实时求解许多实际问题, 而且它们的离散实现能以较小的计算量提供问题的较好解, 并为建立和理解数值稳定的算法提供理论基础.

本项目研究了交集上变分不等式的神经网络模型及相关的理论与算法问题. 通过将解的条件转化为新的双射影方程组，建立了求解约束变分不等式问题的有效神经网络模型，构造了多个能量函数, 给了确保模型及其特例稳定和收敛的易于验证的充分条件及其在图像融合中的应用；通过引入新向量, 设计了求解l1-范数优化问题一个简单神经网络，证明了该模型的稳定性和收敛性, 并给出了其在图像恢复中的应用；构造了恰当的能量函数，分别给出了确保一个单层神经网络和几类时滞神经网络稳定、指数稳定的充分条件；分别提出了解图像恢复问题、奇异鞍点问题和线性方程组的几种有效算法，得到了它们的收敛性结果和比较结果；深入研究了时滞离散捕食系统、具有时滞和接种的幼年染病单种群模型、带有疾病和分段常数变量的捕食-被捕食系统和时滞非线性纵向飞行模型等的动态特性，以及噪声和捕捞对捕食生态系统稳定性的影响、乘性信号和非高斯噪声激励下FHN神经系统的随机共振及相关性影响、基因选择模型中有界噪声和时间延迟诱导的相变等；分别设计了解多目标优化和复杂优化问题的三种进化算法，以及几种量子签名及秘密共享方案. 建立的神经网络模型复杂性低且稳定性好，能实时求解许多实际问题, 且它们的离散实现能以较小的计算量提供问题的较好解. 所提模型和方法为求解相关实际问题提供了新途径，所得结果为它们的进一步应用与研究提供了必要的理论基础.