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Physics informed deep learning for computational elastodynamics without labeled data
arXiv - CS - Computational Engineering, Finance, and Science Pub Date : 2020-06-10 , DOI: arxiv-2006.08472
Chengping Rao and Hao Sun and Yang Liu

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.

中文翻译:

物理学为没有标记数据的计算弹性动力学提供深度学习

在过去的几十年中,通过求解控制偏微分方程 (PDE) 对固体力学问题进行建模的数值方法(如有限元)一直在蓬勃发展。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场。物理信息深度学习是最近开发的一种新方法,用于对 PDE 解决方案进行建模,并有望在不使用任何标记数据的情况下解决计算力学问题。其背后的原理是通过深度神经网络 (DNN) 来逼近感兴趣的数量(例如,PDE 解变量)并嵌入物理定律来规范网络。为此,训练网络相当于最小化一个精心设计的损失函数,其中包含 PDE 残差和初始/边界条件 (I/BC)。在本文中,我们提出了一个具有混合变量输出的物理信息神经网络 (PINN) 来模拟弹性动力学问题,而无需求助于标记数据,其中 I/BC 几乎没有强加。特别是,受混合有限元分析的启发,将位移和应力分量都作为 DNN 输出,这在很大程度上提高了网络的准确性和可训练性。由于传统的 PINN 框架使用拉格朗日乘法器以“软”方式增加了所有剩余损失分量,因此弱强加的 I/BC 不能很好地满足,尤其是当存在复杂的 I/BC 时。为了克服这个问题,基于多个单个 DNN 建立了 DNN 的复合方案,以便可以以“硬”的方式强制满足 I/BC。提议的 PINN 框架在具有不同 I/BC 的几个数值弹性示例上得到了证明,包括静态和动态问题以及截断域中的波传播。结果显示了 PINN 在计算力学应用中的前景。
更新日期:2020-06-16
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