论文标题
物理学告知计算弹性动力学的深度学习，没有标记数据
Physics informed deep learning for computational elastodynamics without labeled data
论文作者
论文摘要
在过去的几十年中，数值方法（例如有限元素）一直在蓬勃发展，用于通过解决局部微分方程（PDE）来建模固体力学问题。区分这些数值方法的一个显着方面是它们如何近似感兴趣的物理领域。物理知识深度学习是一种用于建模PDE解决方案的新方法，并显示出在不使用任何标记数据的情况下解决计算机械问题的希望。其背后的哲学是通过深神经网络（DNN）近似关注数量（例如PDE解决方案变量），并嵌入物理定律以使网络正常。为此，训练网络等同于最大程度地限制了精心设计的损耗函数，该损失函数包含PDE残差和初始/边界条件（I/BCS）。在本文中，我们提出了一个具有混合变量输出的物理信息的神经网络（PINN），以模拟弹性动力学问题，而无需诉诸标记的数据，其中几乎不施加I/BC。特别是，位移和应力组件都被视为DNN输出，灵感来自混合有限元分析，这在很大程度上可以提高网络的准确性和训练性。由于常规的Pinn框架以Lagrange乘数以“软”方式增强了所有残差损失组件，因此弱施加的I/BC无法得到充分满足，尤其是在存在复杂的I/BC时。为了克服这个问题，基于多个单个DNN建立了DNN的综合方案，以便可以以“硬”方式强行满足I/BC。在具有不同I/BC的几个数值弹性示例上，提出的PINN框架已证明，包括静态和动态问题，以及截短域中的波传播。结果显示了PINN在计算技工应用的背景下的希望。
Numerical methods such as finite element have been flourishing in the past decades for modeling solid mechanics problems via solving governing partial differential equations (PDEs). A salient aspect that distinguishes these numerical methods is how they approximate the physical fields of interest. Physics-informed deep learning is a novel approach recently developed for modeling PDE solutions and shows promise to solve computational mechanics problems without using any labeled data. The philosophy behind it is to approximate the quantity of interest (e.g., PDE solution variables) by a deep neural network (DNN) and embed the physical law to regularize the network. To this end, training the network is equivalent to minimization of a well-designed loss function that contains the PDE residuals and initial/boundary conditions (I/BCs). In this paper, we present a physics-informed neural network (PINN) with mixed-variable output to model elastodynamics problems without resort to labeled data, in which the I/BCs are hardly imposed. In particular, both the displacement and stress components are taken as the DNN output, inspired by the hybrid finite element analysis, which largely improves the accuracy and trainability of the network. Since the conventional PINN framework augments all the residual loss components in a "soft" manner with Lagrange multipliers, the weakly imposed I/BCs cannot not be well satisfied especially when complex I/BCs are present. To overcome this issue, a composite scheme of DNNs is established based on multiple single DNNs such that the I/BCs can be satisfied forcibly in a "hard" manner. The propose PINN framework is demonstrated on several numerical elasticity examples with different I/BCs, including both static and dynamic problems as well as wave propagation in truncated domains. Results show the promise of PINN in the context of computational mechanics applications.
