采用两步优化器的深度配点法与深度能量法求解薄板弯曲问题  

The Application of Deep Collocation Method and Deep Energy Method with a Two-step Optimizer in the Bending Analysis of Kirchhoff Thin Plate

作　　者：郭宏伟 庄晓莹[2,3] Hongwei Guo;Xiaoying Zhuang(Chair of Computational Science and Simulation Technology,Institute of Photonics,Faculty of Mathematics and Physics,Leibniz University Hannover,Hannover 30167,Germany;Department of Geotechnical Engineering,Tongji University,Shanghai,200092;Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education,Tongji University,Shanghai,200092)

机构地区：[1]戈特弗里德•威廉•莱布尼茨汉诺威大学,汉诺威.德国,30167 [2]同济大学土木工程学院地下建筑与工程系,上海200092 [3]同济大学岩土及地下工程教育部重点实验室,上海200092

出　　处：《固体力学学报》2021年第3期249-266,共18页Chinese Journal of Solid Mechanics

摘　　要：随着计算机技术的进步以及机器学习算法的进一步发展,深度学习方法逐渐被广泛应用于各行各业中.论文发展并比较了适应于工程计算的深度配点法与深度能量法并应用于求解薄板弯曲问题.深度配点法采用物理信息驱动的深度神经网络,通过将物理信息(偏微分方程强形式)引入到损失函数中,最终将求解薄板弯曲问题简化为优化问题.深度能量法则是采用系统总势能驱动的神经网络.根据最小势能原理,在所有的可能位移场中,真实位移场的总势能取最小值,因此我们可以使用总势能构造损失函数,从而求解薄板弯曲问题.对于边界条件,通过罚函数法将有约束最优化问题转化为求解无约束最优化问题.深度配点法与深度能量法的适用性基于神经网络的通用近似定理.由于物理信息跟总势能的引入,增加了神经网络训练的困难,为了解决这个问题,我们发展了两步优化器方法.数值结果表明,深度配点法与深度能量法很适合求解薄板弯曲问题,并且程序实现简单,实现了真正意义上的"无网格法".With the advancement of computing power and machine learning algorithms,deep learning methods have been widely applied in a wide range of fields.In this manuscript,we develop the deep collocation method and the deep energy method fitted to engineering computation and further apply them to solve the Kirchhoff thin plate bending problems.The deep collocation method adopts the physics-informed neural networks,incorporating the strong-form governing equations into the loss function.It reduces the solving of thin plate problem into an optimization problem.On the other hand,the deep energy method utilizes energy-driven neural networks based on the principle of minimum potential energy,indicating that of all displacements satisfying given boundary and equilibrium conditions,the actual displacement is the one that minimizes the total potential energy at stable equilibrium.Thus,we can build a loss function from the total potential energy.With the boundary conditions penalized to the loss form,the problem is reduced to an unconstrained optimization one.The physics-informed and energy-driven neural networks are based on the universal approximation theorem.Due to the introduction of physical and energy information,the neural networks become difficult to train.An improved two-step optimization algorithm is presented to train the neural network.From the numerical results,it is clearly seen that both methods are suitable for solving thin plate bending problems,easy to implement,and truly"meshfree".

关 键 词：深度学习 配点法 能量法 薄板弯曲 偏微分方程 

分 类 号：TB12[理学—工程力学]