大规模稀疏线性系统的神经网络算法研究

Most problems in the scientific and engineering computing have to be transformed into linear problems to be solved numerically. The study for numerical solutions to linear systems is mainly focused on seeking efficient numerical iterative methods. Nowadays, with the hardware and software of computer highly developed and the information storing almost not a problem, it is hard to get great improvement for traditional numerical methods. Meanwhile, information processing and computer scientists are working on constructing new generation of computer which is closer to the function of the human brain based on artificial neural network theory. Therefore, it is particularly necessary to study the neural network algorithm for solving linear systems numerically. The project intends to study the neural network algorithm for large sparse linear systems, including two aspects: First, according to the characteristics of linear equations, design effective and efficient neural network algorithms, which includes improvements for the existing methods in literatures and designing new algorithms according to the latest achievements of the neural network algorithms (such as feedforward neural network learning algorithm) and analyze their stability; Secondly, study for the selection of the initial weights and determination of the best learning rate of the designed neural network algorithms, which is the key issue to be resolved, like to determine the optimal parameters of the traditional numerical methods, it is very challenging. With the above work, we hope to establish a relatively systematic neural network algorithm theory for large sparse linear systems.

科学工程计算中的大部分问题最终都要转化为线性问题进行数值求解，对线性系统数值方法的研究主要集中在寻求高效的数值迭代方法上，在计算机软硬件高度发达、信息存储几乎不成问题的今天，传统的数值方法很难取得质的改进，而信息处理和计算机科学家正致力于根据人工神经网络理论构造更加逼近人脑功能的新一代计算机，所以为线性系统的数值求解问题设计研究相应的神经网络算法就显得尤为必要。本项目拟对大型稀疏线性系统的神经网络算法进行研究，主要包括两方面的内容：首先，根据线性方程组自身的特点，设计构造高效的神经网络算法，这包括对现有文献中方法的改进和根据神经网络算法的最新成果（比如前馈神经网络学习算法）设计全新的算法并分析其稳定性；其次，对设计的神经网络算法的初始权值选取和最佳学习率确定进行研究，这也是拟解决的关键问题，与传统数值方法的最优参数确定一样，是很有挑战的。通过对已有成果改进、创新，以形成相对系统的算法理论。

项目对大型稀疏线性系统的神经网络算法进行了研究，主要包括数值积分、数值微分、线性方程组、矩阵方程和变线性矩阵不等式求解的神经网络算法模型，讨论了算法初始权值和最佳学习率的选取范围，形成了相关的算法理论。美中不足的是没有与传统的数值方法，比如子空间方法及预条件技术进行比较，这些可以作为后续工作以使理论更加完善。