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.

本文提出了一种基于物理学信息神经网络（PINN）的方法来解决在没有标注数据的情况下建模弹性动力学问题的挑战，进一步解决了复杂的 I/BCs 在弱正则化 PINN 框架下无法很好满足的问题，在多个数值弹性例子中展示了该方法的可行性。