薄板弯曲问题的神经网络方法

为发展神经网络方法在求解薄板弯曲问题中的应用，基于Kirchhoff板理论，提出一种采用全连接层求解薄板弯曲四阶偏微分控制方程的神经网络方法。首先在求解域、边界中随机生成数据点作为神经网络输入层的参数，由前向传播系统求出预测解；其次计算预测解在域内及边界处的误差，利用反向传播系统优化神经网络系统的计算参数；最后，不断训练神经网络使误差收敛，从而得到薄板弯曲的挠度精确解。以不同边界、荷载条件的三角形、椭圆形、矩形薄板为例，利用所提方法求解其偏微分方程，与理论解或有限元解对比，讨论了影响神经网络方法收敛的因素。研究表明，本文方法对求解薄板弯曲问题的四阶偏微分控制方程具有一定的适应性，其收敛性受多种条件影响。相比有限元，该方法收敛速度较慢，在复杂的边界条件下收敛性不佳，然而其不基于变分原理，无需计算刚度矩阵，便可获得较高的计算精度。

NEURAL NETWORK METHOD FOR THIN PLATE BENDING PROBLEM

Recently, deep learning has made good progress in various disciplines. In order to develop the application of deep learning technology in solid mechanics, a neural network method with fully connected layers is proposed to solve the Kirchhoff thin plate bending problems which is governed by the fourth-order partial differential equation (PDE). Firstly, the training points from domain and boundary are randomly generated and feed into the forward propagation system of neural network to obtain the prediction solution. Then the errors are calculated by the loss function proposed in this paper. The parameters inside neural network are then optimized by the back propagation system. Finally, the neural network is trained continuously to make the errors converge, and the deflection solution of thin plate bending is then obtained. Taking triangle, ellipse and rectangular thin plates with different boundary and load conditions as examples, the partial differential equation is solved by the method proposed, and the results are compared with the theoretical solution or finite element method solution. In the end, the factors affecting the convergence of the neural network method are studied. It is found that the method is capable of solving the fourth order partial differential equations of thin plate bending problems. The convergence of this method is affected by the boundary conditions, optimization algorithms, numbers of hidden layers and neurons, and the chosen of learning rate. Compared to finite element method, the neural network method faces the problem of slow convergence speed. However, it is not based on the variational principle. It can obtain high accuracy without the calculation of stiffness matrix. The solution domain is discretized by the randomly generated points. The neural network method is flexible and can also be treated as meshless method. It can provide new ideas in the research of large deformation and nonlinear problems in the future.