理性神经网络方法及其在粘弹性谱分析中的应用

The neural network computing can become an efficient and real-time structural analysis method. However, due to the bad generalized ability of networks there still exist some problems such as the computational precision,the reliability of results, and so on. The ill-posedness of the above problems is the most fundamental reason for the bad generalized ability. In this project, the mechanical problem is turned into the boundary value problem of mathematical and physical equations. Then, a rational neural network computing method for the structural analysis is proposed. a neural network computing model is constructed. Firstly, the error energy function of the neural network is determined according to the mathematical and physical equations. Furthermore, the functions of the network topology and neurons activations are derived by the essential condition of the extreme value of Tikhonov functional analysis. Secondly, the number and distribution of the training samples and the number of neurons in the hidden layers are determined according to the computational accuracy of the neural network.Thirdly, the improved error back propagation algorithm is adopted to train the samples which are the mixed boundary conditions. In view of the above, a rational neural network computing method based on the structural analysis is developed. Finally, the response to the spectrum analysis of random vibration for solid rocket motor is taken as a research content and the viscoelastic fractional order derivative model for propellant is established. Combined the method proposed in this project with the correspondence principle method and the dynamic substructure method,the results of spectrum analysis are obtained. The effectiveness of the present method can be verified by comparing with the results of the finite element method and the experimental mechanics method.

神经网络计算有望成为一种高效乃至实时的结构分析方法，但是由于网络泛化能力差的原因，其在计算精度、结果的可靠性等方面仍存在诸多亟待解决的问题。其中，不适定性是泛化能力差的根本原因。本课题拟将力学问题归结为数理方程的边值问题，并以此为先导建立一种结构分析的理性神经网络计算方法。首先在网络误差能量函数上，将数理方程与神经网络融合，由Tikhonov泛函取极值的必要条件导出网络拓扑与神经元激活函数；其次以网络建模精度为依据，确定样本数量、分布及隐层神经元个数；然后以边界条件作为网络学习样本，通过研究针对混合边界条件的全局收敛训练算法，实现结构分析的神经网络建模；最后以固体火箭发动机随机振动问题为研究对象，建立药柱粘弹性分数阶导数模型，将所提出的理性神经网络方法与对应原理、动态子结构法相结合得到谱分析结果。与有限元法及试验结果进行对比，以验证所提方法的有效性。

神经网络计算有望成为一种高效乃至实时的结构分析方法，但是由于网络是一个黑箱系统以及泛化能力差等原因，其在计算精度、结果的可靠性和如何解释网络参数的物理意义等方面仍存在诸多亟待解决的问题。针对上述问题，项目组展开了深入的研究，主要包括三方面的内容。一是在利用已有力学知识（控制方程）构建神经网络模型方面，首先给出了用于积分计算的对偶神经网络方法，解决了传统方法求解多重积分难的问题。然后将对偶神经网络计算的应用范围从积分问题发展为可同时包括积分、微分算子的情况，导出了对应偏微分方程（组）及其边界条件的多个对偶神经网络结构形式及其训练样本。由此将偏微分方程（组）求解问题映射为一个网络参数相互关联的多神经网络训练问题。以各神经网络性能函数之和作为神经网络系统的性能函数，结合Levenberg-Marquardt优化算法给出了一种多神经网络的联合训练算法，以完成神经网络系统对偏微分方程解的拟合。通过上述系列工作，实现了偏微分方程（组）的理性神经网络求解。将弹性力学控制方程（组）作为偏微分方程（组）的具体形式，初步形成了弹性力学问题的理性神经网络求解理论体系。二是在基于神经网络的粘弹性材料参数拟合方面，利用自定义神经网络模型对粘弹性材料广义Maxwell模型的松弛模量进行拟合。根据Prony级数每一项的具体形式来确定神经网络中相应隐层单元的激活函数类型，神经元的个数与Prony级数的项数相对应。训练收敛后的神经网络既实现了对材料参数的拟合又给出了神经网络参数的物理意义。此研究为神经网络参数的物理意义表征和参数识别提供了一种有效途径。三是在理性神经网络的工程应用方面，利用上述所提方法，给出了基于直接积分法的对偶神经网络结构可靠度计算、求解弹性力学平面应力问题以及粘弹性梁的随机振动谱分析等应用范例。